University of Alberta · controllers using accurate models of the processes. Modelling and control of ﬁxed-bed reactors have been the topic of many researches since early 70’s

University of Alberta

Optimal Control of Fixed-Bed Reactors with Catalyst Deactivation

by

Leily Mohammadi Sardroud

A thesis submitted to the Faculty of Graduate Studies and Research in partialfulfillment of the requirements for the degree of

Doctor of Philosophy.

in

Chemical Engineering

Department of Chemical and Materials Engineering

c�Leily Mohammadi Sardroud.Spring 2013

Edmonton, Alberta

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reproduced in any material form whatever without the author’s prior written permission.

To my beloved parents · · ·

Bu tezi sevgili annem ve babama ithaf ediyorum · · ·

Abstract

Catalytic reactors have widespread applications in chemical and petrochemical

industries. The most well known type of catalytic reactors are fixed-bed or packed-bed

reactors where the reaction takes place on the surface of the catalyst.

One of the most important phenomena that takes place in a catalytic fixed-

bed reactor is catalyst deactivation. Catalyst deactivation can have variety of

consequences. It can have negative e↵ects on the conversion and selectivity of the

desired reaction. Consequently, it will a↵ect the productivity and energy e�ciency

of the plant. It is therefore important to design e�cient controllers that are able

to track the optimal pre-defined trajectories of the operating conditions to ensure

optimal operation of the plant.

Depending on the transport and reaction phenomena occuring in a fixed-bed

reactor, it can be modelled by a set of partial di↵erential equations (PDEs) or a mixed

set of PDEs and ordinary di↵erential equations (ODEs). Moreover, the governing

transport phenomena (i.e. di↵usion or convection) dictates the type of PDEs involved

in the model of the reactor (i.e. parabolic or hyperbolic).

In this work, infinite dimensional optimal control of a fixed-bed reactor with

catalyst deactivation is studied. Since dynamical properties of hyperbolic PDEs

and parabolic PDEs are completely di↵erent, they are discussed as di↵erent topics.

The thesis begins with optimal control of a class of fixed bed reactors with catalyst

deactivation modelled by time-varying hyperbolic equations. Then the model

predictive control of this class of distributed parameter systems under parameter

uncertainty is explored. The optimal control of reactors modelled by parabolic PDEs

is first explored for the case of reactors without catalyst deactivation. Then the

proposed controller is extended to a more general class of distributed parameter

systems modelled by coupled parabolic PDE-ODE systems which can represent fixed-

bed reactors with the rate of deactivation modelled by a set of ODEs.

Numerical simulations are performed for formulated optimal controllers and their

performance is studied.

Acknowledgements

I would like to express my appreciation to my supervisor, Dr. Fraser Forbes, who

encouraged and guided me through all stages of my research. His support and

helpful suggestions were invaluable basis for successful completion of my research.

His excellent supervision provided me with the opportunity to learn how to think

critically, solve problems e�ciently and conduct research independently.

I would also like to express my gratitude to my co-supervisor, Dr. Stevan Dubljevic,

for always being available and challenging me to do better.

I would like to thank Dr. Michel Perrier, Dr. Sirish Shah, and Dr. William

McCa↵rey, who kindly reviewed and evaluated this thesis. I would also like to thank

Dr. Ilyasse Aksikas for the technical discussions and help he provided throughout my

research.

I am grateful to the wonderful friends who supported me emotionally throughout

this program and made this journey much easier: Natalia Marcos, Sahar Navid

Ghasemi, Vida Moayedi, Sarah Nobari, Negin & Negar Razavilar, Maryam & Leila

Zargarzadeh, Garrett & Rayna Beaubien. My special appreciation goes to Padideh G.

Mohseni and Amir Alizadeh who played the role of my family when I was thousands

of kilometres away from them. My research was also benefited from Padideh and

Amir’s technical discussions.

Most important of all, I would like to thank my family in Iran for their

unconditional love and support. Their faith in my capabilities has always been the

greatest source of strength in all stages of my life .

Contents

1 Introduction 1

1.1 Control of distributed parameter systems . . . . . . . . . . . . . . . . 4

1.2 Scope and outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 LQ Control of Time-Varying Hyperbolic Systems 9

2.1 Time-Varying Hyperbolic DPS: Background . . . . . . . . . . . . . . 11

2.2 Problem Statement: Time-Varying Hyperbolic PDEs . . . . . . . . . 14

2.3 Optimal Control Design . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.4 Case study: A Fixed-Bed Hydrotreater . . . . . . . . . . . . . . . . . 18

2.4.1 Trajectory and stability analysis . . . . . . . . . . . . . . . . . 22

2.5 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3 Robust Constrained MPC of Coupled Hyperbolic Systems 34

3.1 Characteristics-Based MPC . . . . . . . . . . . . . . . . . . . . . . . 38

3.2 Robust Characteristic-Based MPC . . . . . . . . . . . . . . . . . . . 46

3.3 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4 LQ Control of Di↵usion-Convection-Reaction Systems 55

4.1 Parabolic DPS: Background . . . . . . . . . . . . . . . . . . . . . . . 58

4.2 LQ Control of Parabolic DPS . . . . . . . . . . . . . . . . . . . . . . 62

4.2.1 Eigenvalue Problem . . . . . . . . . . . . . . . . . . . . . . . . 67

4.3 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.4 Case Study: A Fixed-Bed Reactor with Axial Dispersion . . . . . . . 77

4.4.1 Model Description . . . . . . . . . . . . . . . . . . . . . . . . 77

4.4.2 Simulation Study . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5 LQ Control of Coupled Parabolic PDE-ODE systems 89

5.1 Coupled PDE-ODE Systems: Mathematical Description . . . . . . . . 91

5.2 Coupled PDE-ODE Operator: Eigenvalue Problem . . . . . . . . . . 97

5.3 Stability Analysis of PDE-ODE Systems . . . . . . . . . . . . . . . . 100

5.4 LQ Control of PDE-ODE Systems . . . . . . . . . . . . . . . . . . . . 102

5.5 Case study: Catalytic Cracking Reactor with Catalyst Deactivation . 103

5.6 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 108

5.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

6 Conclusions and Recommendations 113

6.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

Bibliography 118

List of Tables

2.1 Model Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28


4.2 Comparison of the l2-norm . . . . . . . . . . . . . . . . . . . . . . . . 86


List of Figures

2.1 Schematic diagram of the hydrotreating reactor . . . . . . . . . . . . 18

2.2 Time-varying LQ-feedback functions �1

and �2

, ↵ = 1 ⇥ 10�2 . . . . 30

2.3 Time-invariant LQ-feedback functions �1

and �2

, ↵ = 1 ⇥ 10�2 . . . 31

2.4 Integral average of the error . . . . . . . . . . . . . . . . . . . . . . . 32

2.5 Manipulated variable, ↵ = 1 ⇥ 10�2 . . . . . . . . . . . . . . . . . . 33

3.1 Calculation of the state variable along the characteristic curves . . . . 40

3.2 Prediction of state variables for a1

/a2

= 0.4 . . . . . . . . . . . . . . 40


/a2

= 0.5 . . . . . . . . . . . . . . 45


/a2

= 0.6 . . . . . . . . . . . . . . 46

3.5 Temperature trajectory under the robust controller . . . . . . . . . . 51

3.6 Concentration trajectory under the robust controller . . . . . . . . . . 52

3.7 Input trajectory under the robust controller . . . . . . . . . . . . . . 53

3.8 Output trajectory under the robust controller . . . . . . . . . . . . . 54

4.1 Approximation of the reactor as a composite media . . . . . . . . . . 69

4.2 Schematic diagram of the catalytic cracking reactor . . . . . . . . . . 78

4.3 Steady state profile of yA and yB . . . . . . . . . . . . . . . . . . . . 82

4.4 Second element of �n given by Equation (4.67) and Equations (4.62)-

(4.63) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.5 Third element of �n given by Equation (4.67) and Equations (4.62)-(4.63) 83

4.6 Closed loop trajectory of yB . . . . . . . . . . . . . . . . . . . . . . . 84

4.7 Closed loop trajectory of the input variable (Inlet Concentration) . . 85

4.8 Closed loop trajectory of yB for finite dimensional controller . . . . . 86

4.9 Trajectory of tracking error . . . . . . . . . . . . . . . . . . . . . . . 87

5.1 Second element of �n given by Equation (4.67) and Equations (4.62)-

(4.63) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

5.2 Third element of �n given by Equation (4.67) and Equations (4.62)-(4.63)110

5.3 Closed loop trajectory of error yB � yBss . . . . . . . . . . . . . . . . 111

5.4 Optimal input trajectory . . . . . . . . . . . . . . . . . . . . . . . . . 112

1Introduction

Catalytic reactors have widespread applications in chemical and petrochemical

industries. They can be regarded as the heart of a chemical plant. The most well

known type of catalytic reactors are fixed-bed or packed-bed reactors, where the

reaction takes place on the surface of the catalyst.

In a chemical plant, commonly the reactors are followed by other unit operations

such as separation processes. Thus, the operation of the downstream processes

strongly depends on the quality of the products from the reactors. Hence, optimal

operation of reactors in each plant is of key economical and operational importance.

One important phenomena that takes place in a catalytic fixed-bed reactor is

catalyst deactivation. Chemical reactions occur on the active sites of the catalyst.

These active sites may be lost for a variety of reasons like coking, poisoning or

sintering. The loss of active sites results in decrease in the activity of the catalyst, as

the activity is proportional to the number of available active sites.

Catalyst deactivation can have variety of consequences. It can have negative

1

2

e↵ects on the conversion and selectivity of the desired reaction. Consequently, it

will a↵ect the productivity and energy e�ciency of the plant. In order to compensate

for the e↵ect of catalyst deactivation, the operating conditions of the reactor are

gradually changed to maintain the selectivity and conversion, which is equivalent to

maintaining the overall quality and rate of production. This is commonly performed

by calculation of optimal trajectories for operating conditions through an o↵-line

dynamic optimization problem. These trajectories are used as the set-point variables

for the controllers. It is therefore important to design e�cient controllers that are

able to track the pre-defined trajectories of the operating conditions.

Temperature is one of the key parameters a↵ecting the reaction rate and selectivity.

An important issue in control of a fixed-bed reactor is to ensure that the reactor

temperature does not exceed the maximum allowable temperature for the catalyst.

In addition to e↵ects on selectivity and conversion, catalyst deactivation may cause

thermal instability of the reactor. Dependence of catalyst activity on concentration

and temperature results in non-uniform catalyst activity in fixed-bed reactors. If

temperature disturbances occur upstream of the reactor, variation in catalyst activity

can result in temperature excursions that are much larger than those in beds of

uniform activity (Jaree et al., 2001; Jaree et al., 2008). The best approach for

temperature control of the fixed-bed reactors is to control the temperature profile

along the length of the reactor. This requires availability of multiple thermocouples

along the length of the reactor and an e�cient controller.

With growing development of advanced control technologies at the industrial scale,

application of model based controllers has become possible. A high-fidelity model of

the process is a key factor for successful implementation of model-based controllers.

Therefore, in the last few decades extensive studies have been performed to develop

controllers using accurate models of the processes.

Modelling and control of fixed-bed reactors have been the topic of many researches

since early 70’s. The earliest models of the fixed-bed reactors were developed by

approximation of a fixed-bed reactor by a sequence of continuous flow stirred tank

3

reactors (CSTR). In this approach the reactor is modelled by a finite number of well-

mixed reactors in series. As the number of the reactors increases, the accuracy of the

model increases, but only an infinite number of interacting CSTRs would be able to

provide an accurate model of a simple fixed-bed reactor.

More sophisticated models of a fixed-bed reactor can be obtained by solving mass

and energy balance equations for the reactor. In order to capture all of the main

“macroscopic” phenomena (i.e., reactions, di↵usion, convection, and so forth), the

model of a fixed-bed reactor takes the form of a set of partial di↵erential equations.

The rate of catalyst deactivation can be modelled by time-varying algebraic equations

or in more complex cases by ordinary di↵erential equations; hence, integration of

catalyst deactivation dynamics into the model of the system results in a set of time-

varying PDEs or a set of coupled PDE-ODE equations. Depending on the governing

transport phenomena in a chemical process, i.e. di↵usion or convection, the partial

di↵erential equations modelling the process may be a set of hyperbolic or parabolic

equations.

In control theory, systems similar to a fixed-bed reactor are regarded as distributed

parameter systems where the dependent variables are functions of time and space;

as opposed to lumped parameter systems where the dependent variables are only

functions of time (e.g., a well-mixed reactor). Other examples of distributed

parameter systems in chemical engineering are fluidized bed reactors, heat exchangers

and reactive distillation columns.

Traditionally, the majority of control methods have been developed for lumped

parameter systems. Thus, the distributed parameter systems are commonly

approximated with lumped parameter systems and controlled using algorithms for

such lumped systems. Due to the widespread existence of DPS and economic benefits

of precise control of these systems, developing controllers that use the most accurate

model of these systems can have substantial performance benefit. Motivated by this,

this thesis is focused on formulation of high performance controllers for distributed

parameter systems with special emphasis on fixed-bed catalytic reactors. In the

Sec. 1.1 Control of distributed parameter systems 4

following sections the available studies on the topic of distributed parameter systems

and the outline of the thesis will be summarized.

1.1 Control of distributed parameter systems

As mentioned before, the most conventional approaches for control of DPS use

lumping techniques to approximate the PDE model of the system by a set of coupled

ODEs. Methods such as finite di↵erences may be used for spatial discretization,

resulting in approximations that consist of ordinary di↵erential equations; however,

such approximations may not be suitable for high performance control schemes, as the

dimensionality of the produced sets of ODEs can be very high when trying to ensure

high performance control. Also controllability and observability of the system will

depend on the number and location of the discretization points (Christofides, 2001).

Another approach for control of DPS is to develop the control algorithm based on

the PDEs and discretize the resulting controller during the implementation. The

advantage of this approach is that the controller synthesis is based on the original

model of the system which represent all the dynamical properties of the system

without any approximation. Over the recent years, research on the control of DPS is

oriented towards development of algorithms that deal with infinite dimensional nature

of these systems.

Mathematical theory of optimal control of distributed parameter systems began in

late 1960s in the work of Lions (1971) and followed by other researchers (e.g., Banks

and Kunisch (1984), Kappel and Salamon (1990) and Curtain and Zwart (1995)).

In a functional analysis framework, systems modelled by PDEs can be formulated

in a state space form similar to that for lumped parameter systems by introducing

a suitable infinite-dimensional space and operators instead of usual matrices in

lumped parameter systems. The type of PDE system (e.g. hyperbolic, parabolic

or elliptic) determines the approach followed for the solution of the control problem

(Christofides, 2001). Based on the governing phenomenon in a chemical process,

convection or di↵usion, the model equations can be hyperbolic or parabolic.


For hyperbolic systems, Callier and J.Winkin (1990)-(1992) studied the LQ control

problem using spectral factorization for the case of finite rank bounded observation

and control operators. Aksikas (2005) extended this approach for the more general

case of exponentially stable linear systems with bounded observation and control

operators. In his work, this method was applied to a nonisothermal plug flow reactor

to regulate the temperature and the reactant concentration in the reactor. The

optimal control problem for a particular class of hyperbolic PDEs is solved via the

well-known Riccati equation, which is developed in a series of papers by Aksikas and

co-workers. (e.g., Aksikas and Forbes (2007), Aksikas et al. (2008a), Aksikas et al.

(2008b), etc). The approach taken in this body of work does not address the case of

time-varying and two time-scale hyperbolic systems, which can model many chemical

engineering processes, including fixed-bed reactors with catalyst deactivation.

In the framework of model predictive control, Shang et al. (2004) studied

characteristic-based model predictive control of hyperbolic systems. The proposed

MPC is shown to have high computation e�ciency without any approximation;

however, input and output constraints are not addressed in this work.

The conventional approach for dealing with parabolic systems is modal

decomposition (Ray, 1981). The main characteristic of parabolic systems is that their

spectrum can be partitioned into a finite slow part and an infinite fast complement.

This feature has been used by many researchers to perform model reduction and

synthesize finite dimensional models representing the distributed parameter system

with arbitrary accuracy (See Christofides (2001), Dubljevic et al. (2005) and references

therein). The low dimensional model is used to formulate predictive controllers

for the infinite dimensional system. Although the model reduction method is

computationally e�cient for di↵usion dominant systems, it can result in a high

dimensional system for convection dominant parabolic systems. Moreover, control

of time-varying infinite dimensional systems as well as coupled systems have not been

explored.

The control action in a DPS can be distributed over the spatial domain of the


system, or can be applied at the boundary. For example, in the case of a fixed-bed

reactor, the manipulated variable can be inlet temperature (i.e., boundary control)

or the temperature profile of the cooling jacket (i.e., distributed control). Boundary

control systems are mathematically more challenging than systems with distributed

control actions. Boundary control of DPS has been explored by few studies. Curtain

and Zwart (1995) and Emirsjlow and Townley (2000) address the transformation of

the boundary control problem into a well-posed abstract control problem. Recently,

Byrnes et al. (2006) studied the boundary feedback control of parabolic systems using

zero dynamics. The proposed algorithm is limited to parabolic systems with self-

adjoint operator A, therefore cannot be used for chemical reactors that are modelled

by multiple PDEs with non-symmetric coupling.

Although the topic of optimal control of infinite dimensional systems is explored

by many researchers, the more complicated case of distributed parameter systems

modelled by a set of coupled PDEs has not been as well studied. Dynamics of a

fixed-bed reactor and most of other chemical processes are modelled by more than one

equation (e.g., mass and energy balances) and these equations are coupled. Available

infinite dimensional controllers are not able to address coupling in the modelling

equations.

As mentioned before, integration of deactivation kinetics into the model of the

reactor leads to a set of time-varying PDEs or a set of coupled PDEs-ODEs. The

research in the area of coupled PDE-ODE systems is relatively scarce and will be

addressed in this thesis.

Motivated by the fact that the precise control of fixed-bed catalytic reactors is of

key importance in chemical industry, the focus of this thesis is infinite dimensional

control of fixed-bed reactors. Each chapter addresses a class of distributed parameter

systems that can represent an industrial fixed-bed reactor. Although the emphasis is

on fixed-bed reactors, the developed controllers are capable of controlling any system

that is modelled by the studied class of distributed parameter systems in each chapter.

Sec. 1.2 Scope and outline 7

1.2 Scope and outline

As mentioned before, catalytic reactors are the most important part of chemical plants

and the optimal control of them has significant economic importance. Integration

of catalyst deactivation in the modelling of the fixed-bed reactor introduces some

complications to the system. The objective of this thesis is to study optimal

control of fixed-bed reactors modelled by a set of partial di↵erential equations.

The main goal is to formulate optimal controllers that are able to mitigate or

ameliorate the e↵ect of catalyst deactivation on reactor performance using the

distributed parameter model of the reactor with no or minimal approximation to

directly formulate the controller. Since hyperbolic and parabolic systems have

completely di↵erent dynamical behaviour and require di↵erent approaches to solve

the control problem, the thesis is divided to two parts. Hyperbolic systems that model

reactors with negligible di↵usion are studied in chapters 2-3 and parabolic systems

modelling di↵usion-convection-reaction systems are discussed in chapters 4-5. The

thesis begins with a simple case of a catalytic reactor with negligible di↵usion and

simple exponentially decaying deactivation kinetics and ends with a complicated case

of di↵usion-convection-reaction system where the catalyst deactivation is represented

by a set of ODEs.

In Chapter 2, it is assumed that the e↵ect of di↵usion is negligible and the system

can be modelled by a set of hyperbolic equations. It is further assumed that the

e↵ect of catalyst deactivation appears as time-varying reactive terms in the PDEs,

resulting in a time-varying infinite dimensional system. This chapter is extension

of the work of Aksikas et al. (2008a), which addresses a less complex case of plug

flow reactors. Conditions on the stabilizability of these systems are studied and

an infinite dimensional LQ controller is formulated to regulate the trajectory of the

output variables around the steady state profile of the system and eliminate the e↵ect

of catalyst deactivation.

Motivated by the fact that the mathematical model of a system always has some

uncertainty, as well as existence of input and output constraints in a plant, in Chapter

Sec. 1.2 Scope and outline 8

3 robust constrained model predictive control of uncertain coupled hyperbolic systems

is studied. It is assumed that the uncertainty in the reactor model arises from

unknown kinetic coe�cients and uncertainty in the inlet flow measurements. In this

work, the system has polytopic uncertainties and a robust MPC using method of

characteristics is developed to ensure satisfaction of input and output constraints

under parameter uncertainty.

Chapter 4 focuses on development of an LQ controller for coupled parabolic systems

modelling a fixed-bed reactor, where both di↵usion and convection phenomena are

significant. An innovative approach is proposed to solve the eigenvalue problem

for coupled parabolic systems with spatially varying coe�cients resulting from

linearization around the steady state profile of the system. Stabilizability of the

system is studied using its spectral properties. Finally, an LQ controller is formulated

and solved using the spectral properties of the system. This chapter is a significant

step in the formulation of a more complicated case for catalytic reactors with catalyst

deactivation modelled by a set of ODEs (i.e., infinite dimensional systems modelled

by coupled PDE-ODE systems), which is developed in Chapter 5.

In Chapter 5, stabilizability and optimal controller formulation for systems

modelled by coupled PDE-ODE systems with in-domain coupling are studied.

Such systems represent fixed-bed reactors with significant di↵usion and convection,

where catalyst deactivation is represented by a set of ODEs. Using several exact

transformations, the set of coupled PDE-ODE system is decoupled and represented

as a Riesz Spectral system. Then, the LQ controller formulated in Chapter 4 is

extended to this form of infinite dimensional systems.

Chapter 6 is the conclusion of the thesis and summarizes the contributions of this

work and possibilities for future work.

2LQ Control of Time-Varying Hyperbolic

Systems

In this chapter the issue of the optimal control of a fixed-bed reactor with negligible

di↵usion and catalyst deactivation is studied. The system is modelled by a set

of coupled time-varying hyperbolic partial di↵erential equations and it is assumed

that the catalyst deactivation is captured through time-varying reaction rates. An

infinite dimensional LQ optimal controller is formulated by solving the corresponding

operator Riccati equation1.

1Portions of this chapter have been published in “Mohammadi, L., I. Aksikas and J. F. Forbes

(2009), Optimal Control of a Catalytic Fixed-Bed Reactor with Catalyst Deactivation. Proceedings

of the American Control Conference”; and submitted to “Mohammadi, L., I. Aksikas, S. Dubljevic,

J. F. Forbes and Y. Belhamadia, Optimal Control of a Catalytic Fixed-Bed Reactor With Catalyst

Deactivation. Journal of Process Control”

9

10

In a catalytic reactor, the catalyst may deactivate during the operation for a variety

of reasons (e.g., poisoning by impurities in feed, formation of coke on catalyst surface,

and so forth). Deactivation of the catalyst can significantly a↵ect the performance

of the reactor. Due to the economic importance of e↵ective operation of catalytic

reactors in chemical plants, formulation of high performance controllers that can

maintain the process at optimal conditions is of crucial importance.

Incorporation of the catalyst deactivation in kinetic modelling of the reaction

results in a time-varying reaction rate. A simplifying assumption in modelling of

a catalytic reactor is existence of negligible di↵usion. Therefore, a catalytic reactor

with negligible di↵usion and decaying catalyst can be modelled by a set of time-

varying hyperbolic PDEs. Many other chemical processes can also be modelled by

this class of infinite dimensional systems (e.g., a counter current heat exchanger with

time-varying heat transfer coe�cient).

The objective of this chapter is to formulate an infinite dimensional state-feedback

LQ controller for systems that are described by a set of time-varying hyperbolic

PDEs. This chapter establishes a baseline for the development of controllers for

more complicated infinite dimensional systems with mixed di↵usion and convection

transport phenomena.

In the recent years, much work has been published on infinite dimensional control

of hyperbolic systems. In the framework of functional analysis, Aksikas et al. (2009)

studied the solution of LQ control problem for hyperbolic systems by solving an

operator Riccati equation. Spectral factorization, which is based on frequency-

domain description, is an alternative method for solving the LQ problem (Callier

and J.Winkin (1990) and Callier and Winkin (1992)). These approaches address only

optimal control of time-invariant hyperbolic systems.

Control of time-varying hyperbolic systems is studied by Aksikas et al. (2008a),

but the approach is limited to plug flow reactors. Models of plug flow reactors consist

of a set of hyperbolic equations with identical convective term. In these systems,

mass and energy balances both have the same time scale. Generally, the model of a

Sec. 2.1 Time-Varying Hyperbolic DPS: Background 11

catalytic reactor consists of multiple PDEs with distinct convective terms and as a

result, the PDEs are multi-time scale set of equations.

The scope of this chapter is time-varying hyperbolic systems with specific focus

on fixed-bed catalytic reactors. A case study of a fixed-bed hydrotreating reactor is

chosen. The goal is to formulate an infinite dimensional LQ controller to regulate

the temperature profile around the desired steady state trajectory and eliminate the

e↵ect of catalyst deactivation.

In this chapter, the approach proposed by Aksikas et al. (2008a) is extended to

the general form of time-varying hyperbolic systems. Also, stability of these systems

is studied using a Lyapunov approach. After formulation of the LQ controller, the

e↵ect of rate of catalyst deactivation on the closed loop performance of the controller

has been investigated through numerical simulations.

The contributions of this chapter are:

• Stability analysis of a general form of linear time-varying hyperbolic systems;

• Solution of LQ optimal control problem for a general form of linear time-varying

hyperbolic systems;

• Implementation of the formulated controller on a fixed-bed reactor.

2.1 Time-Varying Hyperbolic DPS: Background

In this section, we will provide some background information on the infinite

dimensional representation of time-varying hyperbolic systems. These concepts will

be used throughout this chapter to formulate the infinite dimensional controller and

also analyze the stability of the time-varying infinite dimensional systems. As an

example, we will begin with a simple homogeneous hyperbolic equation. It will be

shown, how a system of PDEs can be formulated as an infinite dimensional system

and the fundamental properties of time-varying infinite dimensional systems will

be presented. Consider the initial value problem given by the following hyperbolic

equation:


@x

@t= �@x

@z+ k(t)x

x(s, z) = xs(z)

x(t, 0) = 0

(2.1)

where x denotes the dependent variable, t is time and z denotes space. s is the initial

time. The solution of PDE (2.1) is given by

x(t, z) =

8<

:xs(z � t) exp(

R z

z�tk(⌫)d⌫)

0(2.2)

By introducing X = L2(0, 1) as the state space and x(., t) = {x(z, t), 0 z 1} as

state variable, and the operator A

A(t)h =dh

dz(2.3)

with the domain

D(A(t)) = {h 2 L2(0, 1)|h is absolutely continuous (a.c.),dh

dz2 L2(0, 1) and h(0) = 0}

(2.4)

Equation (2.1) can be represented as an abstract infinite dimensional system

x(t) = A(t)x(t)

x(s) = xs

(2.5)

The infinite dimensional representation of the solution (2.2) can be written as

x(t, z) = (U(t, s)xs)(z) = exp(

Z z

z�t

k(⌫)d⌫)xs(z � t) (2.6)

U(t, s) is the solution operator of the initial value problem (2.1) and is a two

parameter family of operators. If the operator A(t) is independent of t, then the two

parameter family of operators U(t, s) reduces to a one parameter family U(t), which

is the semigroup generated by A.

Evolution system theory is the generalization of the concept of the semigroup

for autonomous operators to the non-autonomous operators, where operator A is a

function of t. For the rest of this section fundamental concepts related to evolution

systems will be introduced.


Definition 2.1.1. (Pazy, 1983, Definition 5.3, p. 129) A two parameter family of

bounded linear operators U(t, s), 0 s t T , on X is called an evolution system

if the following two conditions are satisfied:

i) U(s, s) = I, U(t, r)U(r, s) = U(t, s) for 0 s r t T

ii) (t, s) ! U(t, s) is strongly continous for 0 s t T

Definition 2.1.2. (Schnaubelt and Weiss, 2010, Definition 2.2) Let {A(t) :

D(A(t)) ! X|t 2 [0,1} be a family of densely defined linear operators on X.

We say that A(.) generates the evolution family U if U(t, s)D(A(t)) ⇢ D(A(t))

for all t, s 2 [0,1] and for every s 2 [0,1] and every x0

2 D(A(t)) the function

x(.) = T (., s)x0

is continuously di↵erentiable and is the unique solution of the Cauchy

problem

x(t) = A(t)x(t), x(s) = x0

, s t < 1 (2.7)

Definition 2.1.3. (Pazy, 1983, Definition 2.1, p. 130) A family of infinitesimal

generators of C0

-semigroups on a Banach spaceX is called stable if there are constants

M � 1 and ! such that

⇢(A(t)) � (!,1) for t 2 [0, T ] (2.8)

and ��

kY

j=1

R(� : A(tj))

�� M(�� !)�k for � > ! (2.9)

for every finite sequence 0 t1

t2

, · · · , tk T . In the equations above, ⇢(A(t)) is

the resolvent set of operator A(t) and R(� : A(tj)) is the resolvent operator defined

by

R(� : A(tj))x =

1Z

0

e��jsStj(s)xds for �j > ! (2.10)

and ⇢(A(t)) is the resolvent set of the operator A(t).

Theorem 2.1.1. (Pazy, 1983, Theorem 2.3, p. 132) Let {A(t)}t2[0,T ]

be a stable

family of infinitesimal generators with stability constants M and !. Let B(t), 0 <

Sec. 2.2 Problem Statement: Time-Varying Hyperbolic PDEs 14

t < T be bounded linear operators on X. If kB(t)k K for all 0 T , then

{A(t)+B(t)}t2[0,T ]

is a stable family of infinitesimal generators with stability constants

! +KM .

Theorem 2.1.2. (Pazy, 1983, Theorem 4.8, p. 145) Let {A(t)}t2[0,T ]

be a stable

family of infinitesimal generators of C0

-semigroups on X. If D(A(t)) = D is

independent of t and for v 2 D, A(t)v is continuously di↵erentiable in X then there

exsits a unique evolution system U(t, s), 0 s t T , satisfying the following

conditions

i) kU(t, s)k Me!(t�s) for 0 s t T

ii) @@tU(t, s)v|t=s = A(s)v for v 2 D, 0 s T

iii) U(t, s)D ⇢ D for 0 s t T

2.2 Problem Statement: Time-Varying

Hyperbolic PDEs

The problem of interest in this chapter is a system that can be modelled by a set of

coupled first-order time-varying hyperbolic PDEs in one dimensional spatial domain.

Such systems can model a fixed-bed reactor with catalyst deactivation and negligible

di↵usion. Since the objective of this chapter is to formulate an LQ controller for such

systems, it is assumed that the PDEs are linearized around the steady state profile

of the system. The mathematical description of this kind of system can be given by

the following equations

@x

@t(z, t) = V

@x

@z(z, t) +M(z, t)x(z, t) +N(z, t)u(z, t)

y(z, t) = S(z, t)x(z, t)(2.11)

with boundary and initial conditions

x(0, t) = xB 8t � 0

x(z, 0) = x0

(z) 8z 2 [0, 1](2.12)

Sec. 2.2 Problem Statement: Time-Varying Hyperbolic PDEs 15

where x(., t) = [x1

(., t), · · · , xn(., t)]T 2 H := L2(0, 1)n denotes the vector of state

variables, z 2 [0, 1] 2 R and t 2 [0,1) denote spatial position and time, respectively.

u(., t) = [u1

(., t), · · · , um(., t)] 2 U := L2(0, 1)m denotes the input variable; y(., t) =

[y1

(., t), · · · , yp(., t)] 2 Y := L2(0, 1)p denotes the output variable. M,N and C are

matrices of appropriate sizes, whose entries are functions in L1([0, 1]⇥ [0,1)). V is

a constant diagonalizable non-positive matrix. xB is a constant column vector and

x0

(z) 2 H. Since equation (2.11) is normally the result of linearization of a set of

nonlinear hyperbolic PDEs, the variables are in deviation form. Therefore, without

loss of generality, it is assumed that xB = 0.

The infinite dimensional description of the model (2.11)-(2.12) in the Hilbert space

H is:

x(t) = A(t)x(t) + B(t)u(t)

x(0) = x0

2 H

y(t) = C(t)x(t)

(2.13)

where A(t) is the family of linear operators defined on the domain

D(A(t)) = {x 2 H : x is a.c.,dx

dz2 H and x(0) = 0} (2.14)

by

A(t) = Vd.

dz+M(z, t).I (2.15)

and the input and output operators B and C are given by

B(t) = N(z, t).I and C(t) = S(z, t).I (2.16)

where I is the identity operator.

The objective is to formulate an infinite dimensional controller for the time-varying

infinite dimensional system of the form (2.13). This will be discussed in the next

section. Note that, here, the only assumption made for the form of matrix V is that

it is diagonalizable. This assumption is true if eigenvalues of matrix V are simple.

In most chemical processes, the matrix V is diagonal. The reason for diagonality

of V is that the convective transport of the state variables are independent of each

Sec. 2.3 Optimal Control Design 16

other. Hence, without loss of generality it will be assumed that matrix V is a diagonal

matrix.

2.3 Optimal Control Design

This section deals with the computation of an LQ-optimal feedback operator for

the system (2.13) by using the corresponding operator Riccati equation. The

objective of LQ-optimal control problem is to find a square integrable control

uo 2 L2[[0,1);L2(0, l)] which minimizes the cost functional

J(x0

, u) =

Z 1

0

(hCx(t), PCx(t)i + hu(t), Ru(t)i)dt (2.17)

for any initial state x0

2 H. In the cost function above, P = P0

.I 2 L(Y ) is a positive

operator and R = R0

.I 2 L(U) is a self-adjoint, coercive operator in L(U), where P0

and R0

are two positive functions.

The LQ control problem is a classical optimal control problem and its solution can

be obtained by finding the positive self-adjoint operator Q 2 L(H) that solves the

operator Riccati di↵erential equation (ORE), viz.

[Q+ A⇤Q+QA+ C⇤PC � QBR�1B⇤Q]x = 0 (2.18)

for all x 2 D(A), where Q(D(A)) ⇢ D(A⇤).

Lemma 2.3.1. (Bensoussan et al., 2007, Theorem 5.2, p.507) Consider the infinite

dimensional system (2.13), with control and observation operators given by (2.16).

Assume that {A(t), B} is exponentially stabilizable. Then the corresponding operator

Riccati equation (2.18) has a nonnegative bounded solution Q and for any initial state

x0

2 H, the quadratic cost (2.17) is minimized by the unique control uo given on t � 0

by

uo(t) = �R�1

0

B⇤Qx(t).

Sec. 2.3 Optimal Control Design 17

Now we are in a position to state the main theorem of this section, which gives an

expression of the optimal state feedback in terms of the solution of a matrix Riccati

partial di↵erential equation:

Theorem 2.3.1. Consider the linear model (2.13). Assume that P0

is a positive

matrix and R0

is a self-adjoint coercive matrix such that � := diag(�1

,�2

, · · · ,�n) is

the solution of the matrix Riccati partial di↵erential equation:

@�

@t= V

@�

@z� M⇤�� M � C⇤

0

P0

C0

+ �B0

R�1

0

B⇤0

�,

�(t, l) = 0, t 2 [0,1](2.19)

Then Q0

:= �(t, z)I is the unique self-adjoint nonnegative solution of the operator

Riccati di↵erential equation.

Proof. Substituting Q0

:= �(t, z)I in the ORE (2.18) leads to:

[�+ A⇤�+ �A+ C⇤PC � �BR�1B⇤�]x = 0 (2.20)

where Ah = V dhdz

+ Mh and A⇤ is the adjoint operator of A and is defined by

A⇤h = �V dhdz

+M⇤h. Substituting A and A⇤ in the above equation results in:

@�

@tx = V

@�x

@z� M⇤�x � �V @x

@z� M�� C⇤PCx+ �BR�1B�x (2.21)

Since@�x

@z= �

@x

@z+@�

@zx (2.22)

equation (2.21) becomes

@�

@tx = V �

@x

@z+ V

@�

@zx � M⇤�x � �V @x

@z� M�� C⇤PCx+ �BR�1B�x (2.23)

Since V and � are both diagonal, V � = �V and the above equation can be simplified

to:@�

@tx = V

@�

@zx � M⇤�x � M�� C⇤PCx+ �BR�1B�x (2.24)

On the other hand, Qo should satisfy Qo(D(A)) ⇢ D(A⇤). This condition is only

valid if �(t, l) = 0 as this implies that �(t, l)x(l) = 0, 8x 2 D(A) and consequently

Qo(D(A)) ⇢ D(A⇤).

Sec. 2.4 Case study: A Fixed-Bed Hydrotreater 18

The condition for the existence and uniqueness of the solution for the Operator

Riccati equation, is exponential stabilizability of infinite dimensional system (2.13).

Exponential stabilizability of this system depends on parameters of the system and

should be studied for each case. In the following sections, a case study will be

considered and its exponential stability will be proven. Finally, the associated matrix

Riccati equation will be solved to find the solution for the optimal control problem.

2.4 Case study: A Fixed-Bed Hydrotreater

The process considered in this chapter is a fixed-bed hydrotreating reactor with

catalyst deactivation. The dynamics of the reactor can be described by partial

di↵erential equations derived from mass and energy balances. To model the reactor,

a plug-flow pseudo-homogeneous model is considered. Moreover, we consider a one-

spatial dimension model where there are no gradients in the radial direction. The

schematic diagram of the process is shown in Figure 2.1

Figure 2.1: Schematic diagram of the hydrotreating reactor

In the simplified system considered here, a lumped reaction kinetics equation was

assumed which has the following form (see (Chen et al., 2001)):

A+H2

! C

rA = k(t)e(�ERT )Cn1

A Cn2H

(2.25)

In the above equation, A represents Naphtha and C represents the products of the

hydrotreating reaction.

Remark 2.4.1. The more accurate kinetic model of the reaction is represented

by Langmuir-Hinshelwood-Hougen-Watson (LHHW) rate equations. Under the


assumption that all adsorption steps are weak, the LHHW rate equation will be

simplified to a power law rate equation.

Given the modeling assumptions, the dynamics of the process are described by the

following energy and mass balance partial di↵erential equations (PDE’s).

✏@CA

@t= �⌫ @CA

@z� ⇢Bk(t)e

� ERT Cn1

A Cn2H

@T

@t= �⌫ @T

@z+⇢B�Hr

⇢Cp

k(t)e�ERT Cn1

A Cn2H

(2.26)

with the boundary conditions given, for t � 0, by:

CA(0, t) = CA,in,

T (0, t) = Tin,(2.27)

The initial conditions are assumed to be given , for 0 z l, by

CA(z, 0) = CA0

(z),

T (z, 0) = T0

(z)(2.28)

In the equations above, CA, T, ✏, ⇢B, ⇢, Cp, E, CH ,�Hr, ⌫ denote the reactant

concentration, the temperature, the porosity of the reactor packing, the catalyst

density, the fluid density, the activation energy, the concentration of hydrogen, the

enthalpy of reaction, and the superficial velocity respectively. In addition, t, z and l

denote the time and space and the length of the reactor, respectively. T0

and CA0

denote the initial temperature and reactant concentration profiles, respectively, such

that T0

(0) = Tin and CA0

(0) = CA,in. k is the pre-exponential factor. As a result of

catalyst deactivation, this coe�cient varies with time. Generally k is a function of

time and operating conditions, but here we assume that the operating conditions are

maintained in narrow ranges. Therefore, k is only a function of time and is assumed

to be given by:

k = k0

+ k1

e�↵t (2.29)

The above expression for kinetics of naphtha hydrotreating reaction is in agreement

with the observations that after a rapid initial deactivation of the hydrotreating

catalyst there is a slow deactivation phase and finally a stabilization of the catalyst

activity phase (Kallinikos et al., 2008).


Since temperature and concentration of the reactor can have di↵erent orders of

magnitude, first we will transform the nonlinear equations to a dimensionless form.

Then the nonlinear equations will be linearized around the steady state profile of

the system and will be represented as an infinite dimensional system. Finally the

exponential stabilizability of the system will be proven and the optimal control

problem will be solved.

Dimensionless model

Let us consider the following state transformation:

✓1

=T � Tin

Tin

, ✓2

=CA,in � CA

CA,in

(2.30)

Then we obtain the following equivalent representation of the model (2.26).

@✓1

@t= �⌫ @✓1

@z+

�h0

+ h1

e�↵t�(1 � ✓

2

)n1 eµ✓11+✓1 (2.31)

@✓2

@t= �⌫

✏

@✓2

@z+

�l0

+ l1

e�↵t�(1 � ✓

2

)n1 eµ✓11+✓1 (2.32)

with the boundary conditions:

✓1

(0, t) = 0, ✓2

(0, t) = 0 (2.33)

where:

µ =E

RTin

, l0,1 =

⇢B✏k0,1C

n2H Cn1

Aine�µ, (2.34)

h0,1 =

⇢B(��H)

⇢CpTin

k0,1C

n2H Cn1

Aine�µ (2.35)

Infinite-dimensional linearized model

Let us denote by ✓ss and uss the dimensionless steady state profile of the model

(2.31)-(2.32) at the operating point. By defining the following state variables:

x(t) = ✓(t) � ✓ss (2.36)


and new input u(t) = ⌫(t) � ⌫ss, the linearization of the system (2.31)-(2.32) around

its operating profile leads to a linear time-varying infinite-dimensional system (2.13)

on the Hilbert space H := L2(0, l)⇥L2(0, l). Operators A(t), B and C are defined in

the following.

Operator A is given by

A(t) = V.d.

dz+M(t, z).I (2.37)

where V and M are defined by:

V :=

"v1

0

0 v2

#(2.38)

where

v1

= �⌫ss, v2

= �⌫ss✏

(2.39)

M(t, z) :=

"m

11

(t, z) m12

(t, z)

m21

(t, z) m22

(t, z)

#(2.40)

where the functions mij are given by:

m11

= µ(h0

+ h1

e�↵t)(1 � ✓

2ss)n1

(1 + ✓1ss)2

eµ✓1ss1+✓1ss ,

m12

= �n1

(h0

+ h1

e�↵t)(1 � ✓2ss)

n1�1eµ✓1ss1+✓1ss ,

m21

= µ(l0

+ l1

e�↵t)(1 � ✓

2ss)n1

(1 + ✓1ss)2

eµ✓1ss1+✓1ss ,

m22

= �n1

(l0

+ l1

e�↵t)(1 � ✓2ss)

n1�1eµ✓1ss1+✓1ss .

The input operator B = B0

.I 2 L(L2(0, l), H) is the linear bounded operator

where:

B0

=

"B

1

B2

#, (2.41)

B1

=@✓

1,ss

@z, B

2

=1

✏

@✓2,ss

@z

and it is assumed that the full state measurement is available along the length of the

reactor, therefore, C = I.


2.4.1 Trajectory and stability analysis

This section is devoted to the trajectory and the exponential stability of the linearized

fixed-bed reactor model described in the previous section. The following theorem

shows the existence and uniqueness of an evolution system generated by the family

of operators {A(t)}0tT , for any T > 0.

Theorem 2.4.1. Let T > 0. Consider the family of operators {A(t)}0tT given by

(2.37) . Then, there exists a unique evolution system UA(·, ·) : {(t, s) 2 R2 : s t T} such that

@

@tUA(t, s)x = A(t)UA(t, s)x,

8x 2 D(A(t)), 0 s t T.

Moreover, there are constants M � 1 and ! such that

kUA(t, s)k Me!(t�s), 0 s t T.

Proof. The operator A(t) can be written as

A(t) = A0

+M(t) =

"�v

1

d.dz

0

0 �v2

d.dz

#+

"m

1

(t, z)I m2

(t, z)I

m3

(t, z)I m4

(t, z)I

#(2.42)

In order to prove this theorem, it su�ces to validate assumptions of Theorem 2.1.2.

The operator A0

is independent of t. Since V < 0, the operator A0

is infinitesimal

generator of an exponentially stable C0

-semigroup. M(t), t � 0 is bounded linear

operator, then, by using Theorem 2.1.1, A(t) is a stable family of infinitesimal

generators. On the other hand the domain of A(t) is independent of time and for any

x 2 D0

, A(t)x is di↵erentiable. Therefore, all of the Theorem 2.1.2 assumptions are

satisfied.

Now we are in a position to state a theorem on the exponential stability of the

linearized model.


Theorem 2.4.2. Consider the family of operators {A(t)}t�0

as in Theorem 2.4.1.

Then {A(t)}t�0

generates an exponentially stable evolution system.

Proof. The su�cient condition for exponential stability is existence of a positive

operator P that satisfies the following operator Lyapunov di↵erential equation

(Pandolfi, 1992).

P + PA(t) + A(t)⇤P +Q = 0 with P (D(A(t))) ⇢ D(A(t)⇤) (2.43)

where,

Q = Q0

I =

"Q

1

Q2

Q2

Q3

#(2.44)

and Qi, i = 1, · · · , 3 are arbitrary functions such that the matrix Q0

is positive semi-

definite matrix. If we assume that the solution is of the form:

P (t) =

"P

1

I 0

0 P2

I

#(2.45)

where I is the identity operator and Pi is a real valued function, it can be shown that

the following identities hold:

PA = PA0

+ PM(t)I and A⇤P = A⇤0

PI +M(t)⇤PI (2.46)

On the other hand it can be shown that

PA0

+ A⇤0

PI = VdPdz

I (2.47)

Substituting Equations (2.46)-(2.47) into Equation (2.43) results in the following

matrix Lyapunov partial di↵erential equation

dPdt

= �VdPdz

� PM(t) � M(t)⇤P � Q0

, P(t, 1) = 0, t 2 [0,1) (2.48)

If one can prove that the solution for matrix Lyapunov PDE (2.48) of the form

P(t, z) =

"P

1

0

0 P2

#exists, the operator P = PI will be the solution for the operator


Lyapunov di↵erential Equation (2.43). Positiveness on P implies that the operator

P is positive as well.

In order to prove existence of a solution for Equation (2.48), the method of

characteristics is used. In this case, there are two characteristic equations as follows

dt

dr= 1, t(0) = t

0

(2.49)

dz

dr= v

1

z(0) = 0 (2.50)

dz

dr= v

2

z(0) = 0 (2.51)

Along the characteristic equations, the following equation is equivalent to the

matrix Lyapunov PDE (2.48)

dPdr

= �M⇤(r � t0

)P � PM(r � t0

) � Q0

, P(1) = 0 (2.52)

By performing a change of variable: r = 1� r, the Equation (2.52) can be converted

to the following Matrix Lyapunov equation:

dPdr

= M⇤(r + t0

)P + PM(r + t0

) +Q0

, P(0) = 0 (2.53)

Equation 2.53 can be expanded to

dP1

dr= P

1

m1

(r + t0

) +m1

(r + t0

)P1

+Q1

(2.54a)

dP2

dr= P

2

m4

(r + t0

) +m4

(r + t0

)P4

+Q3

(2.54b)

P1

m2

(r + t0

) +m3

(r + t0

)P2

+Q2

= 0 (2.54c)

Assuming that Q1

, Q3

> 0, Equations (2.54a) and (2.54b) have positive solutions

given by:

P1

(r) =

rZ

0

e

sR

0m1(⌧+t0)d⌧

Q1

e

sR

0m1(⌧+t0)d⌧

ds (2.55)

P2

(r) =

rZ

0

eR s0 m4(⌧+t0)d⌧Q

3

eR s0 m4(⌧+t0)d⌧ds (2.56)


Since Q0

is an arbitrary matrix, one can assume that it has the following form:

Q0

=

"Q

1

�(P1

m2

(r + t0

) +m3

(r + t0

)P2

)

�(P1

m2

(r + t0

) +m3

(r + t0

)P2

) Q3

#

(2.57)

If we can prove that the resulting matrix Q0

is non-negative, then it can be

concluded that the family of operators {A(t)}t�0

generates an exponential stable

evolution system. For positiveness of Q0

, the following condition should hold:

8x 2 D(A(t)), hx,Q0

xi > 0 (2.58)

or

h"

x1

x2

#,

"Q

1

x1

+Q2

x2

Q2

x1

+Q3

x2

#i > 0 (2.59)

which results in

hx1

, Q1

x1

i + hx1

, Q2

x2

i + hx2

, Q2

x1

i + hx2

, Q3

x2

i > 0 (2.60)

Since Q1

and Q3

are assumed to be positive, hx1

, Q1

x1

i and hx2

, Q3

x2

i are both

positive. If one chooses Q1

and Q2

such that Q1

>> Q2

and Q3

>> Q2

, the condition

for positiveness of Q0

holds. This proves that positive operators Q and P satisfying

the dynamic Lyapunov equation (2.43) exist and therefore {A(t)}t�0

generates an

exponentially stable evolution system.

Optimal Control Formulation

In this section the solution of the LQ controller of §2.3 is discussed for the

hydrotreating reactor with infinite dimensional model given by (2.13) and operators

A and B defined by equations (2.37) and (2.41). It is assumed that full state

measurement is available and C = I.

The solution of the LQ controller can be computed by solving the equivalent

matrix Riccati partial di↵erential equation given by equation (2.19). Considering

that M,V,B are given by (2.40), (2.38), (2.41), respectively, then the matrix Riccati

equation (2.19) can be written as a set of partial di↵erential and algebraic equations

Sec. 2.5 Numerical Simulations 26

given as follows (see Aksikas et al. (2009, Comment 3.1) and Aksikas and Forbes

(2007, Comment 3.1)):

@�1

@t= �⌫

1

d�1

dz+ 2m

11

�1

+ c11

� b11

�2

1

,

@�2

@t= �⌫

2

d�2

dz+ 2m

22

�2

+ c22

� b22

�2

2

,

0 = m21

�2

+ �1

m12

+ c12

� �1

b12

�2

,

�1

(t, l) = 0,

�2

(t, l) = 0

(2.61)

where the functions mij, cij and bij are the entries of the matrices M,C⇤0

P0

C0

and

B0

R�1

0

B⇤0

, respectively.

In the equations (2.61), there exist two unknown functions �1

,�2

. Note that there

are three equations. In general, when there exists n state variables, the number of

unknown functions is n, while the number of equations is n(n + 1)/2. Thus the

set of equations (2.61) cannot be solved to find a unique solution; however, if we

assume that some entries of weighting matrices P0

and R0

are unknown, the number

of unknown variables can be made to be equal to the number of equations and the

set of equations solved to find a unique solution for the matrix Riccati eqution. In

general, to complete the number of unknown variables, n(n � 1)/2 entries of P0

or

R0

should be assumed to be unknown. After solving the set of equations, one should

validate the positive definiteness of the weighting function.

2.5 Numerical Simulations

In this section, the performance of the proposed approach is demonstrated. Values

of the model parameters are given in Table 2.1.

The LQ-controller discussed in the previous section was studied via a simulation

that used a nonlinear model of the reactor given in Equations (2.26)-(2.28). The

performance of the proposed controller was compared to an infinite dimensional LQ

controller designed by assuming constant catalyst activity. This LQ controller was


computed using the method proposed in (Aksikas et al., 2009). The solution of LQ

controller based on the time-invariant model of the reactor can be computed by solving

a set of ODEs instead of PDEs given by (2.61). Therefore, it needs less computation

e↵ort than the LQ controller proposed in this work.

The control objective is to regulate the temperature trajectory at the desired

steady state profile. Using the nominal operating conditions and the model given in

Equations (2.26)-(2.28), the steady-state profiles of temperature and concentration

were computed. Then, the nonlinear model was linearized around the stationary

states and transformed to the infinite dimensional form of Equation (2.13). The

feedback functions �1

and �2

are computed by solving a set of equations given by

(2.61). To compute the feedback functions for the second case, it is assumed that the

reaction coe�cient is equal to k0

+k1

during the operation time. Both controllers are

applied to the original nonlinear model of the system given by Equations (2.26)-(2.28).

Simulations are performed using COMSOL R�. For handling possible numerical

instability resulting from discontinuity in the boundary/initial conditions of the

hyperbolic equations, the artificial di↵usion option of the solver is enabled. In order to

investigate the e↵ect of deactivation rate, two di↵erent values of ↵ are considered. In

the first case, ↵ = 1⇥ 10�2 the deactivation time has the same order of magnitude as

the residence time of the system, but in the second case, ↵ = 1⇥10�4, the deactivation

time is much longer than the residence time of the reactor. The feedback functions

�1

and �2

of two controllers for ↵ = 1 ⇥ 10�2 are shown in Figures 2.2-2.3. For

↵ = 1⇥10�4, the feedback functions of time-varying controller have the similar trend

but di↵erent values; however, the feedback functions of time-invariant controller are

not functions of ↵, as the deactivation is ignored for developing this controller.

To assess the performance of two controllers, it is assumed that the system is

initially at steady state and we are interested in maintaining the temperature profile

at that steady state and eliminate the e↵ect of catalyst deactivation. The integral

average of the temperature error is calculated for each case and is shown in Figure 2.4.

From Figure 2.4, one can observe that for ↵ = 1⇥ 10�2, the performance of the time-


Table 2.1: Model ParametersParameter Values unit

✏ 0.4

⇢B 700 kgcat/m3

CH 587.4437 mol/m3

n1

1.12

n2

0.85

E 81000 J/mol

R 8.314 J/mol K

CA0

0.419344 mol/m3

CAin 0.419344 mol/m3

T0

523 K

Tin 523 K

⇢ 2.7 Kg/m3

Cp 147.49 J/Kg K

�H 101.3⇥ 103 J/mol

k0

1.2384

k1

2.8896

L 0.15 m

Sec. 2.6 Summary 29

varying controller is much better than the time-invariant controller. The time-varying

controller, regulates the temperature at the desired steady state profile with almost

no error; however, the time-invariant controller results in oscillatory response with

significant error. As the catalyst deactivation rate decreases the di↵erence between

performance of two controllers decreases. For ↵ = 1⇥10�4, the time-varying controller

is still better than time invariant one, but the maximum temperature error is less than

1K which is not significant. In Figure 2.5, the trajectory of manipulated variable for

two controllers are compared. It can be observed that, although the time-varying

controller has significantly better performance for ↵ = 1 ⇥ 10�2, it does not require

an aggressive input change.

2.6 Summary

In this chapter, an LQ-optimal controller for a time-varying set of two time-scale

coupled hyperbolic equations was formulated. Exponential stability of these systems

was analyzed using Lyapunov equation. It was shown that the solution of the

optimal control problem can be found by solving an equivalent matrix Riccati partial

di↵erential equation. Numerical simulations were performed to evaluate the closed

loop performance of the formulated controller on a hydrotrating fixed-bed reactor.

The performance of the proposed controller was compared to the performance of

an infinite dimensional controller formulated by ignoring the catalyst deactivation.

Simulation results showed that the performance of the proposed controller is better

than the controller ignoring the catalyst deactivation, when the deactivation time is

comparable with resident time of the reactor.

Sec. 2.6 Summary 30

(a) �1

(b) �2

Figure 2.2: Time-varying LQ-feedback functions �1

and �2

, ↵ = 1 ⇥ 10�2

Sec. 2.6 Summary 31

0 0.05 0.1 0.15!0.05

!0.04

!0.03

!0.02

!0.01

0

0.01

0.02

0.03

z (m)

!1

!1

!2

Figure 2.3: Time-invariant LQ-feedback functions �1

and �2

, ↵ = 1 ⇥ 10�2

Sec. 2.6 Summary 32

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000!12

!10

!8

!6

!4

!2

0

2

4

t (s)

1 L

R L 0(T

�T

ss)

dz

LQ controller using time!varying model

LQ controller using time!invariant model

(a) ↵ = 1 ⇥ 10�2

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000!0.3

!0.2

!0.1

0

0.1

0.2

t (s)

1 L

R L 0(T

�T

ss)

dz



(b) ↵ = 1 ⇥ 10�4

Figure 2.4: Integral average of the error

Sec. 2.6 Summary 33

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000.2

0.4

0.6

0.8

1

1.2

t (s)

u/u

ss



Figure 2.5: Manipulated variable, ↵ = 1 ⇥ 10�2

3Robust Constrained MPC of Coupled

Hyperbolic Systems

In this chapter Model Predictive Control of the fixed-bed hydrotreating reactor

introduced in Chapter 2 is studied. LQ optimal controller introduced in Chapter

2 is not suitable to control processes with input/output constraints. Moreover,

LQ controller’s performance may deteriorate under parameter uncertainty. The

robust constrained characteristic MPC is formulated for the general case of

two time-scale hyperbolic system and then its performance is studied for the

case of the hydrotreating reactor. The formulated MPC controller ensures

that the input and output constraints are satisfied under parameter uncertainty1

1Portiones of this chapter have been published in “Mohammadi, L., S. Dubljevic and J. F. Forbes

(2010), Robust Characteristic- Based MPC of a Fixed-Bed Reactor. Proceedings of the American

Control Conference”.

34

35

A mathematical model of a process can never represent dynamical aspects of the

process completely. There is always some sort of uncertainty involved in modelling

a system. A typical source of model uncertainty in a chemical process is unknown

or partially known parameters like reaction rate. Presence of uncertain variables and

unmodeled dynamics, if not taken into account in the controller design, may lead

to poor performance of the controller or even to closed-loop instability. Another

important issue, which should be considered when designing a controller, is existence

of input and/or output constraints. The constraints can represent actuator limitation

or safety requirements. For example, catalytic reactors are very sensitive to changes

in temperature. High temperatures can accelerate the rate of catalyst deactivation;

on the other hand, there is normally a minimum temperature required for the reactor

to operate. When constraints on states and controls are present, in addition to robust

stability, it is necessary to ensure the constraint satisfaction under model parameter

uncertainty

Model Predictive Control is a class of model based controllers, which make explicit

use of a model of the process to obtain the control action by minimizing an objective

function (Camacho, 1999). It is a very well known approach for dealing with

parameter uncertainty and constraints. The objective function in MPC is a measure

of predicted performance of the system over prediction horizon. In other words, by

using a model of the process, MPC predicts the future behavior of the process and

determines the current control action. The control input is updated at every sampling

time (Rossiter, 2003). One of the major advantages of MPC is its ability to do on-

line constraint handling in a systematic way. Another advantage of MPC is that it

can be easily applied to multivariable systems (Camacho, 1999). Since the essence of

MPC is to optimize the prediction of the process behaviour based on a process model

over values for the manipulated variables, the model is a critical element of a MPC

controller (Rawlings, 2000).

Motivated by the fact that robustness of the controller and constraint handling

are very important aspects of controller design, this chapter goes beyond the infinite

36

dimensional LQ controller introduced in Chapter 2 with the objective of development

of a robust MPC controller, which is able to handle input and output constraints

under parameter uncertainty.

Unfortunately, model predictive control algorithms for distributed parameter

systems are relatively scarce. For Di↵usion reaction systems, which are described

by parabolic PDEs, Dubljevic et al. (2005) used modal decomposition to derive

finite-dimensional systems that capture the dominant dynamics of the original

PDE. The low dimensional model is subsequently used for controller design. For

convection dominated parabolic PDE modal decomposition methods result in high-

order finite dimensional systems. Model predictive control for high-order systems is

computational demanding and cannot be applied on-line.

For hyperbolic systems, the eigenvalues of the spatial di↵erential operator cluster

along vertical or nearly vertical asymptotes in the complex plane (Christofides, 2001),

and modal decomposition methods may not be used. Then, Dubljevic et al.(Dubljevic

et al., 2005) used finite di↵erence method to convert the hyperbolic equations to a

set of ODEs and Model Predictive Controller is designed for the resulting model.

As discussed previously, using discretization methods ignores the distributed nature

of the system and dynamics of the system may not be captured properly. One of

the features of hyperbolic systems is that any discontinuity in the initial or boundary

conditions can propagate and finite di↵erence or finite element algorithms may become

numerically unstable, and therefore are not suitable to use as a model for controller

design. Moreover, hyperbolic systems have finite impulse response (FIR) behavior

(Choi and Lee, 2005) and the discretization algorithms do not preserve this property.

The method of characteristic is a classical algorithm for solution of a hyperbolic

equation that can preserve the FIR property of the hyperbolic systems.

Characteristic-based model predictive control is an approach for model predictive

control of DPS that uses method of characteristics combined with model predictive

controller and is proposed by Shang et al. (2004). The characteristic-based MPC

allows controller design for linear, quasilinear, and nonlinear low dimensional PDEs.

37

In this method, partial di↵erential equations are transformed to a set of ordinary

di↵erential equations along characteristic curves, which exactly describe the original

DPS. Then the controller design can be performed on ODEs instead of PDEs without

approximation. Unfortunately, the model predictive control for hyperbolic systems

investigated by Shang et al. (2004) or Shang et al. (2007) did not consider constraints.

In this work the problem of characteristic-based robust constrained MPC of two

time-scale hyperbolic systems is studied. For designing the robust controller, the plant

is assumed to lie in a polytopic set resulting from possible parameter uncertainties.

The objective of robust control design is to ensure that some performance specification

is met by the designed controller, as long as the plant remains within the polytopic

set (Maciejowski (2002) and Kothare et al. (1996)).

In the original characteristic-based MPC proposed by Shang et al. (2004), the

system is realized as an input-output system. In this work, we have reformulated

the prediction method such that the resulting system is state space system that can

handle state constraints. It has been shown that the resulting state space system has

a certain structural features that can be exploited in the formulation of the robust

model predictive controller.

To formulate the robust MPC, we assume that the uncertainties are polytopic. The

robust model predictive controller is realized as a quadratic optimization problem. It

is assumed that the objective of robust controller is to ensure that the constraints

are satisfied as long as the plant lies inside the polytope. Therefore, the quadratic

objective function is defined based on the nominal model of the system. To ensure

constraint satisfaction under parameter uncertainty, the evolution of the constraints

over the prediction horizon under possible parameter uncertainties is used as the

constraint for the quadratic problem.

The case study considered in this chapter is the hydrotreating reactor introduced

in Chapter 2; however, in this chapter it is assumed that the rate of deactivation is

unknown and therefore the pre-exponential factor in the kinetic term is uncertain.

Another uncertainty arises from fluctuations in the inlet flowrate. The proposed

Sec. 3.1 Characteristics-Based MPC 38

robust constrained MPC is applied to the process and the closed loop performance is

analyzed.

The contributions of this chapter can be summarized as:

• State Space realization of two time-scale hyperbolic systems using the method

of characteristics;

• Formulation of stability guaranteed robust model predictive controller that

ensures constraint satisfaction under parameter uncertainty.

3.1 Characteristics-Based MPC

The method of characteristics is a technique for solving hyperbolic partial di↵erential

equations (Shang et al., 2004). The idea behind the method of characteristics is

that every hyperbolic PDE has a characteristic curve associated with it, along which

dynamics evolve, and as a result, the hyperbolic PDE can be represented as an

equivalent system of ODEs.

Consider a quasilinear system of first-order equations with two dependent variables

⌫1

, ⌫2

and two independent variables t and z.

@⌫1

@t+ a

1

@⌫1

@z= f

1

(⌫1

, ⌫2

, u)

@⌫2

@t+ a

2

@⌫2

@z= f

2

(⌫1

, ⌫2

, u) (3.1)

If coe�cients a1

6= a2

, the system has two di↵erent characteristics determined by:

Characteristic C1

:dz

dt= a

1

Characteristic C2

:dz

dt= a

2

(3.2)

along these characteristics dynamic of the system is described by the following set of

ODEs:

d⌫1

dt= f

1

(⌫1

, ⌫2

, u) along characteristic C1

d⌫2

dt= f

2

(⌫1

, ⌫2

, u) along characteristic C2

(3.3)


Then, by using the method of characteristics, the set of partial di↵erential equations

(3.1) is transformed to a set of ODEs along the characteristic curves. This set of ODEs

can be used to predict the system’s evolution. The characteristic ODEs are coupled

with respect to the two characteristic curves, and the values of state variables at any

point in time and space should be determined by simultaneous integration of both

characteristic ODEs along two nonparallel characteristic curves passing through that

point. This can be used for evaluation of the future values of state variables. In Fig.

3.1 the calculation of the future output variables using the method of characteristics is

illustrated. This method for prediction of the future behaviour is proposed in (Shang

et al., 2004). The idea is that each point in space and time (e.g. P ) has a domain of

dependence. The domain of dependence for point P in Figure 3.1 is the area enclosed

by characteristic curves passing through point P. The solution at P depends on the

values of state variables within its domain of dependence. In other words, the solution

at P can be calculated by integration of characteristic equations along characteristic

curves P � Q and P � R. Values of state variables at Q and R are used as initial

conditions for integration. The mathematical description is

⌫1

(P ) =

Z t(P )

t(Q)

f1

(Q)dt (3.4)

⌫2

(P ) =

Z t(P )

t(R)

f2

(R)dt (3.5)

where:

t(P ) =a1

t(Q) � 2a2

t(Q) + a2

t(R) + Z(R) � Z(Q)

a1

� a2

(3.6)

The position of the point P is calculated by:

Z(P ) =a1

Z(R) � a2

Z(Q) + a1

a2

[t(R) � t(Q)]

a1

� a2

(3.7)


Z (Space)

t (T

ime)

C1

C2

RQ

P

Figure 3.1: Calculation of the state variable along the characteristic curves

Space

Tim

e

3! t

3! t

2 31 4

1 2 3 4

1 2 3

431

4

2

! t! t2

! t1

Figure 3.2: Prediction of state variables for a1

/a2

= 0.4


State Space Representation

In this chapter we are interested in robust model predictive control of linear systems,

therefore before attempting to derive the state space model of the system, assume

that the model equations (3.1) are linearized about the steady state profile of the

reactor. The linearized model can be represented by:

@⌫ 01

@t= �a

1

@⌫ 01

@z+M

11

(z)⌫ 01

+M12

(z)⌫ 02

+ B1

(z)u0

@⌫ 02

@t= �a

2

@⌫ 02

@z+M

21

(z)⌫ 01

+M22

(z)⌫ 02

+ B2

(z)u0 (3.8)

where 0 indicates that the variables are in deviation form.

The characteristic curves for the linearized model (3.8) are:

C1

=dz

dt= a

1

(3.9)

C2

=dz

dt= a

2

(3.10)

and the characteristic equations are:

d⌫ 01

dt= M

11

⌫ 01

+M12

⌫ 02

+ B1

u0 (3.11)

d⌫ 02

dt= M

21

⌫ 01

+M22

⌫ 02

+ B2

u0 (3.12)

Using the prediction method discussed in the previous section, values of ⌫ 01

and ⌫ 02

can be calculated along the length of the reactor. At any time t = tk, the length of the

reactor can be discretized using a finite number of discretization point. By defining

the values of ⌫1

and ⌫2

at discretization points as state variables, the set of PDEs can

be represented as a state space model. The idea is that at t = tk the measurements

of the state variables are available at the discretization points. These measurements

are used to determine the value of the state variables at the intersections of the

characteristic curves passing the initial discretization points. Using the approach

discussed in the previous section, the values of the state variables can be calculated

at the intersection points. A graphical representation of this method is shown in

Figures 3.2-3.4. The state vector is defined as:


x =h⌫ 011

⌫ 012

· · · ⌫ 01N ⌫ 0

21

⌫ 022

· · · ⌫ 02N

iT

where N is the number of discretization points. The values of state variables at

time T = �t can be calculated using Equations (3.4)-(3.5). This procedure will be

repeated for T = 2�t, 3�t, . . . Since the positions of the intersection points change

at each sample time, the resulting state space model is a linear time-varying discrete

system represented by

xk+1

= Akxk +Buk (3.13)

yk = Ckxk (3.14)

The positions of the intersection points are functions of the ratio of slopes of two

characteristic curves, and therefore the structure of A depends on the ratio of two

slopes.

Example 3.1.1. As an example, we will explore the structure of matrix A for the

case that this ratio is equal to 0.4. In Figure 3.2, positions of the intersections are

illustrated at di↵erent sample times. If one repeats the integration procedure for six

sample times, it will be observed that the structure of the matrix A is repeated every

third sample time. The reason is that the positions of the intersection points also

change periodically with period of three sample times. The structure of the matrix

A is given by

A1

=

2

666666666666664

0 0 0 0 0 0 0 0

↵11

(1) 0 0 0 ↵12

(1) 0 0 0

0 ↵11

(2) 0 0 0 ↵12

(2) 0 0

0 0 ↵11

(3) 0 0 0 ↵12

(3) 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

↵21

(1) 0 0 0 ↵22

(1) 0 0 0

0 ↵21

(2) 0 0 0 ↵22

(2) 0 0

3

777777777777775


A3

=

2

666666666666664

↵11

(1) 0 0 0 ↵12

(1) 0 0 0

0 ↵11

(2) 0 0 0 ↵12

(2) 0 0

0 0 ↵11

(3) 0 0 0 ↵12

(3) 0

0 0 0 ↵11

(4) 0 0 0 ↵12

(4)

0 0 0 0 0 0 0 0

↵21

(1) 0 0 0 ↵22

(1) 0 0 0

0 ↵21

(2) 0 0 0 ↵22

(2) 0 0

0 0 ↵21

(3) 0 0 0 ↵22

(3) 0

3

777777777777775

A2

= A1

and Ak =

8>>><

>>>:

A1

k = 3n+ 1

A2

k = 3n+ 2

A3

k = 3n

where ↵ij(l) is the result of the integration of the characteristic equations given by:

↵11

(l) =

Z tk

tk�1

M11

(l)dt

↵12

(l) =

Z tk

tk�1

M12

(l)dt

↵21

(l) =

Z tk

tk�1

M21

(l)dt

↵22

(l) =

Z tk

tk�1

M22

(l)dt

(3.15)

and l represents the spatial position of each discretization point.

The structure of the matrix B also depends on the ratio of two slopes and for the


case of 0.4 is given by

B1

=

2

666666666666664

B1

(1)�t

B1

(2)�t

B1

(3)�t

B1

(4)�t

B2

(1) ⇥ (�t ��t2

)

B2

(2)�t

B2

(3)�t

B2

(4)�t

3

777777777777775

, B2

=

2

666666666666664

B1

(1) ⇥ (2�t ��t1

)

B1

(2)�t

B1

(3)�t

B1

(4)�t

B2

(1) ⇥ (2�t � 3�t2

)

B2

(2) ⇥ (2�t � 2�t2

)

B2

(3)�t

B2

(4)�t

3

777777777777775

B3

=

2

666666666666664

B1

(1)�t

B1

(2)�t

B1

(3)�t

B1

(4)�t

B2

(1) ⇥ (3�t � 4�t2

)

B2

(2)�t

B2

(3)�t

B2

(4)�t

3

777777777777775

, Bk =

8>>><

>>>:

B1

k = 3n+ 1

B2

k = 3n+ 2

B3

k = 3n

(3.16)

Matrix C depends on the available measurements. Assuming that the only output

variable is the value of ⌫1

at z = l, output matrix C will be:

C1

=h0 0 0 1 +M

11

(4) ⇥ (�t1

��t) 0 0 0 M12

(4) ⇥ (�t1

��t)i

C2

= [], C3

=h0 0 0 1 0 0 0 0

i

Ck =

8>>><

>>>:

C1

k = 3n+ 1

C2

k = 3n+ 2

C3

k = 3n

(3.17)

⇤

As mentioned, the position of the intersection points, and therefore the structure

of the state space system, is a function of the ratio of the slopes of two characteristic

curves. In Figures 3.3-3.4, the intersection points are illustrated for the cases that

the ratio is equal to 0.5 and 0.6, respectively. It can be observed that for the first


case, the position of intersection points are constant. Therefore, this case will result

in a linear time-invariant state space system. In the second case, the period is equal

to two, which means that the resulting state space system will have a period of two

sample times.

Space

Tim

e

Space

Tim

e

! t

! t

1 2 3 4

4321

1 2 3 4


/a2

= 0.5

In order to use common MPC algorithms, the periodic system should be converted

to a linear time-invariant system. If the period of the system is N , this can be done

by assembling N sample times to one larger sample time and defining a new state

space system with larger sample time. For the case that the period is equal to 3, the

system can be converted to a linear time-invariant system by defining

A = A3

A2

A1

(3.18)

B =hA

3

A2

B1

A3

B2

B3

i(3.19)

C = [C1

, C2

A1

, C3

A2

A1

]T (3.20)

Finally, the LTI representation suitable for the design of MPC controller is given as:

xk+1

= Axk + Buk, (3.21)

yk = Cxk,

Sec. 3.2 Robust Characteristic-Based MPC 46

Space

Tim

e

Space

Tim

e

2! t

2! t

! t

1 2 3 4

4321

1 2 3 4


/a2

= 0.6

Now that we successfully derived a linear time-invariant state space system, we can

proceed with designing a robust model predictive controller that will be addressed in

the next section.

3.2 Robust Characteristic-Based MPC

In order to include the parametric uncertainty of the plant, it is assumed that the

small variations in the uncertain parameters of the system generate a family of the

linear systems given by Equation (3.21). It is assumed that the set of uncertain models

generated by the uncertain parameters, [Ai Bi] build a polytopic set ⌦{[Ai Bi]}and the real model of the system lies within or on the convex hull of the polytopic

set. Thus the system can be described by a polytopic uncertain system. Polytope ⌦

is defined by

⌦ = Co{[A1

B1

], [A2

B2

], · · · , [AL BL]} (3.22)

So if [A, B] 2 ⌦ then, for some nonnegative �1

,�2

, · · · ,�L summing to one, we


have

[A B] =LX

i=1

�i[Ai Bi]

which means that there is a convex hull of uncertain plants, which is generated by

considering all possible combinations of uncertain parameters.

In this work the model predictive control algorithm proposed by Muske and

Rawlings (1993) is modified to consider parameter uncertainty. It is assumed that

the objective of the robust controller is only to ensure constraint satisfaction under

any possible parameter uncertainty. Hence, the quadratic performance index is

constructed based on the nominal model of the system, while the construction of

the constraints captures the dynamics of all possible parametric uncertain models.

Matrix A in Equation (3.21) is a nilpotent matrix, which implies that the system

has an FIR property. This FIR property is a natural feature of co-current hyperbolic

systems and implies that the system is always stable and all eigenvalues of A are inside

the unit circle. In the case of a tubular reactor, the reason for the FIR property is

that each species is processed in a limited time in the reactor and leaves out of the

reactor in finite time. Therefore, an impulse change in reactor input will propagate

only for finite time.

For stable systems, Muske and Rawlings (1993) proposed the following stability

guaranteed MPC formulation for a LTI state space systemP

(A,B,C) at time k

minuN

yTk+NQyk+N +N�1X

j=0

(yTk+jQyk+j + uTk+jRuk+j +�uT

k+jS�uk+j) (3.23)

Subject to:

x(k + 1) = Ax(k) + Bu(k) (3.24)

y(k) = Cx(k) (3.25)

umin uk+j umax, j = 0, 1, . . . , N � 1 (3.26)

ymin yk+j ymax, j = 0, 1, . . . , N � 1 (3.27)

�umin �uk+j �umax, j = 0, 1, . . . , N � 1 (3.28)


In the cost function (3.23), Q is the weighting on the final value of the state

variables. In order to ensure the stability of the closed loop system, Q should be the

solution of the following Lyapunov equation (Muske and Rawlings, 1993)

Q = CTQC + AT QA (3.29)

Since we are interested in the robust constraint satisfaction, the objective function

used in this work is similar to Equation (3.23), but the constraints should change to

consider all the possible realizations of the system. The constraints that should be

included in the MPC problem are

x(k + 1) = Aix(k) + Biu(k) (3.30)

y(k) = Cix(k) i = 1, 2, ..., L (3.31)

umin uk+j umax, j=0,1,...,N�1

(3.32)

ymin yk+j ymax, j = 0, 1, . . . , N � 1 (3.33)

�umin �uk+j �umax, j = 0, 1, . . . , N � 1 (3.34)

The MPC problem can be converted to the following convex quadratic problem that

can be solved by any quadratic programming algorithm (Muske and Rawlings, 1993).

minuN�k = (uN)THuN + 2(uN)T (Gxk � Fuk�1

) (3.35)

Subject to:2

66666666666666666666666664

I

�I

A1

A2

...

AL

�A1

�A2

...

�AL

W

�W

3

77777777777777777777777775

uN

2

66666666666666666666666664

i1

i2

B1

B2

...

BL

D1

D2

...

DL

w1

w2

3

77777777777777777777777775


Matrices, H,G and F are calculated by determining the prediction of the output

variable as a function of the input variable and are given by

H =

2

66664

BT QB +R + 2S BTAT QB � S · · · BTATN�1QB

BT QAB � S BT QB +R + 2S · · · BTATN�2QB

......

. . . · · ·BT QAN�1B BT QAN�2B · · · BT QB +R + 2S

3

77775(3.36)

G =

2

66664

BT QA

BT QA2

...

BT QAN

3

77775, F =

2

66664

S

0...

0

3

77775(3.37)

Matrices Ai,W, i1

, i2

,Bi are defined to consider the constraints over the prediction

horizon and under possible realizations of the system and are defined as

Ai =

2

66666664

CiBi 0 0 0

CiAiBi CiBi 0 0

CiA2

iBi CiAiBi CiBi 0...

......

...

CiAN�1

i Bi CiAN�2

i Bi CiAiiN�3Bi CiBi

3

77777775

, i = 1, . . . , L (3.38)

Bi =

2

66666664

ymax � CiAi

ymax � CiA2

i

ymax � CiA3

i...

ymax � CiANi

3

77777775

, Di =

2

66666664

�ymin + CiAi

�ymin + CiA2

i

�ymin + CiA3

i...

�ymin + CiANi

3

77777775

(3.39)

W =

2

66666664

1 0 0 0

�1 1 · · · 0...

. . . . . ....

0 · · · �1 1

0 0 0 �1

3

77777775

(3.40)

w1

=

2

6666664

�umax + uk�1

�umax

�umax

�umax

�umax

3

7777775, w

1

=

2

6666664

�umin � uk�1

�umin

�umin

�umin

�umin

3

7777775(3.41)

Sec. 3.3 Case Study 50

The resulting quadratic problem can be solved with any quadratic programming

algorithm, and as long as the real plant lies inside the polytope (3.22) the formulated

MPC is able to handle constraints.

3.3 Case Study

In this section, the proposed characteristic based MPC is applied to the hydrotreating

reactor that was introduced in Chapter 2, and its closed loop performance is analyzed.

The objective is to control the reactor’s outlet temperature at a specified setpoint by

manipulating the superficial velocity of the reactor. It is assumed that the parameters

k and vss are uncertain variables. Therefore, the linearized real model of the system

lies in the convex hull of a polytopic set with four vertices. It is assumed that the pre-

exponential factor, k, can have 40% uncertainty and vss can have 20% uncertainty.

It is also assumed that the manipulating and output variables have lower and upper

constraints that shall not be violated.

The state space model that is used for formulating the controller is constructed

using the proposed method in §3.1. In order to develop the model for controller

formulation, 20 discretization points are used. For simulation of the real plant 60

discretization points are used to construct the model and the controller is applied to

this model.

The prediction horizon is assumed to be 20 sample times. It should be mentioned

that the resulting state space system is a FIR system, so increasing the prediction

horizon to more than resident time of the reactor will not improve the performance

of the controller.

To evaluate the closed loop performance of the controller, it is assumed that the

system is not initially at steady state and the initial temperature and concentrations

are: T0

= 0.95Tin, CA0 = CAin . A random model which lies inside the polytopic set

is chosen as the real model of the system. Numerical simulations are performed to

evaluate the performance of the controller. To solve the quadratic problem, Matlab’s

quadratic programming tool is used.

Sec. 3.3 Case Study 51

Figures 3.5 and 3.6 illustrate the profiles of the reactor temperature and

concentration respectively. Figure 3.7 shows the profile of the manipulated variable

and Figure 3.8 is the trajectory of the control variable.

As Figures 3.7-3.8 show the constraints on input and output trajectories are

satisfied even if the real model of the system is not one of the models considered

during the controller formulation.

Note that the chosen initial and boundary conditions have discontinuity (See

Figures 3.5 and 3.6), however, the proposed proposed algorithm is numerically stable

and the discontinuity only propagates along the characteristic curve.

0

50

100

0

0.05

0.1

0.15400

500

600

700

Time (min)Space (m)

Te

mp

era

ture

(K

)

Figure 3.5: Temperature trajectory under the robust controller

Sec. 3.4 Summary 52

0

50

100

0

0.05

0.1

0.15

0

0.2

0.4

0.6

0.8

Time (min)Space (m)

Co

nc

en

tra

tio

n (

mo

l /

m3)

Figure 3.6: Concentration trajectory under the robust controller

3.4 Summary

In this chapter a characteristic based robust model predictive control algorithm

was developed for systems modelled by two time-scale coupled hyperbolic partial

di↵erential equations. The set of partial di↵erential equations was transformed to

a set of ODEs using the method of characteristics. In order to construct the state

space model of the system, the spatial domain was discretized to a finite number

of intervals. The structure of the resulting state space model was explored. It was

shown that the resulting state space system has certain periodic features depending

on the ratio of slopes of two characteristic curves. It was also shown that the proposed

method preserved the finite impulse response property of the co-current hyperbolic

system. This ensures that the dynamics of the system is captured properly, unlike

other discretization algorithms that cannot preserve this property and may result in

Sec. 3.4 Summary 53

0 20 40 60 80 100 120

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

Time (min)

u /

uss

Figure 3.7: Input trajectory under the robust controller

numerical instability of the model if any discontinuity exists.

In order to deal with parameter uncertainty, it was assumed that the uncertain

models of the system generate a polytopic set and the real model of the system lies

in the polytope ⌦. The MPC problem was defined by nominal performance index,

but the evolution of the constraints over the prediction horizon was considered for

all polytopic models. This formulation of robust controller ensures that constraints

are satisfied as long as the real model of the plant lies inside the polytope; however,

the performance of the controller may deteriorate as only the nominal model on the

system is considered in the performance index.

A case study involving a hydrotreating reactor was used to evaluate the closed loop

response of the controller. In this case study two uncertain parameters existed which

lead to four polytopic models. Simulation results indicated that this algorithm is able

to satisfy input and output constraints under parameter uncertainty.

Sec. 3.4 Summary 54

0 20 40 60 80 100 120300

350

400

450

500

550

600

650

700

750

Time (min)

Ou

tle

t te

mp

era

ture

(K

)

Figure 3.8: Output trajectory under the robust controller

4LQ Control of

Di↵usion-Convection-Reaction Systems

This chapter addresses the optimal control of fixed-bed reactors modelled by a

set of coupled parabolic PDEs. This chapter takes the first step in solution of

the optimal control problem for coupled parabolic PDE-ODE systems, which can

represent fixed-bed reactors with catalyst deactivation modelled by a set of ODEs.

In this chapter an innovative approach is proposed to solve the eigenvalue problem

for a set of coupled parabolic PDEs with spatially varying coe�cients. This class

of parabolic PDEs is the result of linearization of the nonlinear model of the system

around the steady state profile of the system. Using the spectral properties of the

system stability of the system is studied and the optimal control problem is solved1.

1Portions of this chapter have been published in “Mohammadi, L., I. Aksikas, S. Dubljevic and

J. F. Forbes (2012a). LQ-boundary control of a di↵usion-convection-reaction system. International

Journal of Control 85, 171-181” and “Mohammadi, L., I. Aksikas, S. Dubljevic and J. F. Forbes

(2011), Linear Quadratic Optimal Boundary Control of a Di↵usion-Convection-Reaction System.

Proceedings of the 18th IFAC World Conference”

55

56

The focus of the previous two chapters was on the optimal control of systems

modelled by a set of hyperbolic PDEs (i.e. catalytic reactors with negligible di↵usion).

In the real world, pure hyperbolic systems are rare. Normally, hyperbolic equations

result from ignoring the e↵ect of di↵usion phenomenon within the system. In most

chemical engineering processes, more specifically in most fixed-bed reactors, di↵usion

may not be ignored and as a result, the model of the system should be represented

by a set of parabolic PDEs.

Motivated by this, the current chapter focuses on the infinite dimensional control

of systems described by a set of parabolic PDEs, which can be used to represent a

fixed-bed reactor with both di↵usion and convection phenomena. The only di↵erence

between a first order hyperbolic PDE and a parabolic PDE is presence of the

second order derivative with respect to the space variable. This di↵erence results

in completely di↵erent dynamical behaviour and mathematical properties. Unlike

hyperbolic systems, for parabolic systems the operator Riccati equation cannot be

converted to a matrix Riccati equation. Therefore, we need an alternative method to

solve the operator Riccati equation. Parabolic systems can be characterized by their

set of eigenvalues and eigenfunctions. The spectral property of these systems is the

tool that is generally used to deal with parabolic systems.

Chemical processes are normally modelled by a set of nonlinear equations and in

order to apply linear controllers, one needs to linearize the system of equations around

the steady state profile of the system. By linearizing the nonlinear equations about the

steady state profile of the system, a set of linear parabolic PDEs with spatially varying

coe�cients is produced. Solution of the eigenvalue problem for a single parabolic

PDE with constant coe�cients is very well known in the mathematical literature, but

there is no general algorithm to solve the eigenvalue problem for a parabolic PDE

with spatially varying coe�cients. In this work, an innovative idea is proposed to

solve the eigenvalue problem of parabolic PDEs with spatially varying coe�cients.

The proposed approach uses the techniques for the solution of the heat equation for

composite media.

57

Control of di↵usion-reaction systems, which are described by parabolic PDEs has

studied by many researchers (e.g., Christofides (2001), Dubljevic et al. (2005) and

and references therein). In these works, modal decomposition is used to derive finite-

dimensional system that captures the dominant dynamics of the original PDE and

is subsequently used for low dimensional model predictive controller design. The

potential drawback of this approach is that for di↵usion-convection-reaction systems

the number of modes that should be used to derive the ODE system may be very large,

which leads to computationally demanding controllers. Additionally, the control

problem for coupled parabolic systems has not been addressed in these studies.

Boundary control of parabolic systems have been explored in few studies. Curtain

and Zwart (1995) and Emirsjlow and Townley (2000) introduced transformation of

the boundary control problem to a well-posed abstract control problem using an exact

transformation. Recently, Byrnes et al. (2006) studied the boundary feedback control

of parabolic systems using zero dynamics. The proposed algorithm is limited to

parabolic systems with self-adjoint operator A, therefore cannot be used for chemical

reactors that are modelled by multiple PDEs with non-symmetric coupling. Moreover,

the proposed controller is not optimal and may result in aggressive control actions.

Model predictive boundary control of parabolic systems is studied by Dubljevic and

Christofides (2006). This work uses the exact transformation introduced by Curtain

and Zwart (1995) along with modal decomposition to solve the boundary control

problem for a parabolic PDE, but it is also limited to systems described by a single

parabolic PDE. To the best of the author’s knowledge, there is no detailed study that

addresses optimal boundary control of spatially varying coupled parabolic PDEs in

general.

In this chapter, we are interested in the boundary control of systems modelled

by coupled parabolic PDEs with spatially varying coe�cients. The boundary

control problem is transformed into a well-posed abstract boundary control problem

by applying an exact transformation similar to the transformation introduced by

(Curtain and Zwart, 1995). The stabilizability of the resulting systems is also

Sec. 4.1 Parabolic DPS: Background 58

investigated. Stability of linear parabolic systems with constant coe�cients is studied

by Winkin et al. (2000), Delattre et al. (2003) for the special case of a tubular reactor

with two state variables. In this chapter, we extend this work for the more general

case of coupled parabolic systems with spatially varying coe�cients and arbitrary

number of state variables. Finally, by using the spectral properties of the system, the

operator Riccati equation is converted into a set of coupled algebraic equations, which

can be solved numerically. In this work, our focus is on parabolic systems defined on

one-dimensional spatial domain, but the approach can easily be extended to a more

general form of parabolic systems with 2D or 3D spatial domain.

The majority of the previous work on infinite dimensional systems modelled by

parabolic PDEs concentrated on cases that are described by a single spatially invariant

parabolic PDE. This work is a first step in investigation of the control techniques for

systems that are described by a set of PDEs with spatially varying coe�cients, which

would model many chemical engineering processes. A case study involving a tubular

reactor, wherein the catalytic cracking of gas oil takes place is used to illustrate our

method.

The main contributions of this chapter can be summarized as:

• Solution of eigenvalue problem for a set of coupled parabolic PDEs with spatially

varying coe�cients;

• Formulation and solution of an optimal boundary control problem for systems

modelled by a set of coupled parabolic PDEs;

• Stability analysis of coupled parabolic systems.

4.1 Parabolic DPS: Background

In this section, we will provide some background for infinite dimensional

representation of parabolic partial di↵erential equations. The introduced concepts

will be used throughout this chapter to formulate the infinite dimensional controller

and analyze the stabilizabily of the parabolic systems. We will use the following heat


equation for illustration

@x

@t(x, t) =

@2x

@z2(x, t), x(z, t) = x

0

(z)

@x

@z(0, t) = 0

@x

@z(1, t) = 0

(4.1)

By introducing X = L2

(0, 1) as the state space and x(., t) = {x(z, t), 0 z 1}as the state variable, and the operator A

Ah =d2h

dz2with

D(A) = {h 2 L2

(0, 1)|h, dhdx

are absolutely continuous,

d2h

dz22 L

2

(0, 1) anddh

dz(0) =

dh

dz(1) = 0}

(4.2)

the abstract formulation of (4.1) can be written as

x(t) = Ax(t)

x(0) = x0

(4.3)

The solution of the above problem is given by

x(z, t) = hx0

, 1i +1X

n=1

2e�n2⇡2thx0

(.), cos(n⇡.)i cos(n⇡.) (4.4)

In (4.4), �n2⇡2, n = 1, · · · ,1 are eigenvalues of the operator A and cos(n⇡z), n =

1, · · · ,1 are the associated eigenfunctions. In general, eigenfunctions and eigenvalues

of an operator A are found by solving the following eigenvalue problem

AV = �V (4.5)

where � and V are the eigenvalues and corresponding eigenfunctions of the operator

A.

By defining the following operator

T (t)x0

(z) = hx0

(z), 1i +1X

n=1

2e�n2⇡2thx0

(z), cos(n⇡z)i cos(n⇡z) (4.6)


the abstract formulation of (4.4) can be written as

x(t) = T (t)x0

(4.7)

T (t), maps the initial condition x0

to the state of the dynamical system x at time t,

and is the generalization of eAt in finite dimensional systems.

The operator A is an example of Riesz-Spectral operators. In the following,

definitions of Riesz- Spectral operator, Resolvent Set and a special case of Riesz-

Spectral operators (i.e. Sturm-Liouville Operator ) will be provided.

Definition 4.1.1. (Curtain and Zwart, 1995) A sequence of vectors {�n, n � 1} in a

Hilbert space Z forms a Riesz basis for Z if the following conditions hold:

(a) span{�n} = Z;

(b) There exist positive constants m and M such that for arbitrary N 2 N and

arbitrary scalars ↵n, n = 1, · · · , N such that

mNX

n=1

|↵n|2 kNX

n=1

↵n�nk2 MNX

n=1

|↵n|2 (4.8)

Definition 4.1.2. (Guo and Zwart, 2001) Operator A is a Riesz-Spectral operator if

the following conditions hold:

(a) A is closed.

(b) Its eigenvalues are isolated and have finite multiplicity.

(c) The corresponding eigenvectors {�n, n � 1} form a Riesz basis in Z.

(d) The closure of {�n, n � 1} is totally disconnected.

Definition 4.1.3. (Curtain and Zwart, 1995) Let A be a closed linear operator on a

normed linear space X. We say that � is in the resolvent set ⇢(A) of A, if (�I �A)�1

exists and is a bounded linear operator on a dense domain of X. (�I �A)�1 is called

the Resolvent of A.


Theorem 4.1.1. (Curtain and Zwart, 1995) Suppose that A is a Riesz-spectral

operator with distinct eigenvalues {�n, n � 1} and corresponding eigenvectors

{�n, n � 1}. Also assume that { n, n � 1} are the eigenvectors of A⇤ such that

h�n, ni = �nm. Then A satisfies:

a. ⇢(A) = {� 2 C| infn�1

|�� n| � 0}, and for � 2 ⇢(A), (�I � A)�1 is given by

(�I � A)�1 =1X

n=1

1

�� nh., ni�n (4.9)

b. A has the representation

Az =1X

n=1

�nhz, ni�n, z 2 D(A) (4.10)

c. A is the infinitesimal generator of a C0

-semigroup if and only if supn�1

Re(�n) < 1and T (t) is given by

T (t) =1X

n=1

e�nth., ni�n (4.11)

d. The growth bound of the semigroup is given by

!0

= inft>0

✓1

2log(kT (t)k

◆= sup

n�1

Re(�n) (4.12)

Definition 4.1.4. (Winkin et al., 2000) Consider the operator A defined on the

domain

D(A) = {f 2 L2(a, b) : f,df

dzabsolutely continuous,

d2f

dz22 L2(a, b), and ↵a

df

dz(a) + �af(a) = 0,↵b

df

dz(b) + �bf(b) = 0}

(4.13)

where a and b are real numbers, (↵a, �a) 6= (0, 0) and (↵b, �b) 6= (0, 0). A is said to

be a Sturm-Liouville operator if

8f 2 D(A),Af =1

⇢(z)

✓d

dz

✓�p(z)

df

dz(z)

◆+ q(z)f(z)

◆(4.14)

Definition 4.1.5. (Winkin et al., 2000) A linear state space systemP

(A,B,C,D)

is called a Sturm-Liouville system, if and only if �A is a Sturm-Liouville operator.

Sec. 4.2 LQ Control of Parabolic DPS 62

4.2 LQ Control of Parabolic DPS

As mentioned in Chapter 2, LQ control is a well-known optimal control algorithm for

linear state space systems of the form

x = Ax+Bu

y = Cx(4.15)

and assumes a performance metric of the form

J(x0

, u) =

Z 1

0

hy(s), y(s)i + hu(s), Ru(s)ids (4.16)

where u(s) and y(s) are the input and output trajectories, respectively, and R is a

self-adjoint, coercive operator in L(U). In order to formulate the LQ optimal control

problem for a parabolic system, the model of the systems should be formulated

as (4.15). In the following, the formulation of the parabolic PDEs as an infinite

dimensional state space system represented by (4.15) will be discussed. As discussed

before, we will assume that the nonlinear model of the system is linearized around

the steady space profile of the system. Therefore, the coe�cients of the reactive term

are spatially varying.

Consider a system described by a set of linear parabolic PDEs in one spatial

dimension:@✓

@t= D

@2✓

@z2� V

@✓

@z+N(z)✓ (4.17)

with the following boundary and initial conditions

D@✓

@z|z=0

= V(✓z=0

� ✓in)

@✓

@z|z=l = 0

✓(z, 0) = ✓0

(4.18)

where ✓(., t) = [✓1

(., t), · · · , ✓n(., t)]T 2 H := L2(0, l)n denotes the vector of state

variables, z 2 [0, l] 2 R and t 2 [0,1) denote position and time, respectively. D,V,N

are matrices of appropriate sizes, whose entries are functions in L1([0, l]⇥ [0,1)). D


and V are constant diagonal matrices and N is a spatially varying triangular matrix.

We also assume that the ✓in is the manipulated variable.

The system of equations (4.18) is non-self adjoint in general. In order to simplify

the controller synthesis, the system of equations (4.18) can be converted to a self-

adjoint di↵usion-reaction system by the following transformation:

x = T✓ = exp(�D�1V

2z)✓ (4.19)

The resulting system is

@x

@t=D

@2x

@z2+ (TN(z)T�1 � VD�1V

4)x

D@x

@z|z=0

=V

2x|z=0

� V ✓in

D@x

@z|z=l =

V

2x|z=l

x(z, 0) = T�1✓(z, 0)

(4.20)

The above transformation can be performed for any set of parabolic PDEs.

Therefore, without loss of generality we assume that for the rest of this work the

model of the system can be described by the following set of parabolic PDEs.

@x

@t= D

@2x

@z2+M(z)x

D@x

@z|z=0

=V

2x|z=0

� u

D@x

@z|z=l =

V

2x|z=l

x(z, 0) = x0

(z)

(4.21)

The system of equations (4.21) can be formulated as an abstract boundary control

problem on the Hilbert space H := L2(0, l)n (Curtain and Zwart, 1995).8>>>>>><

>>>>>>:

dx(t)dt

= Ax(t),

x(0) = x0

Bx(t) = u(t)

y(t) = Cx(t)

(4.22)


Where A is a linear operator on the domain:

D(A) = {x 2 H : x anddx

dzare a.c. ,

d2x

dz22 H,D

@x

@z|z=l =

V

2x|z=l} (4.23)

(where a.c. means that x is absolutely continuous) and is defined by

A = Dd2.

dz2+M(z).I (4.24)

B : H ! R is the boundary operator with the domain

D(B) = {x 2 H : x is a.c. ,dx

dz2 H} (4.25)

and defined by

Bx(.) = �Ddx(.)

dz|z=0

+V

2x(.)|z=0

(4.26)

and the output operator, C, is defined by the available measurement.

The infinite dimensional system described by (4.22), is not a well-posed infinite

dimensional system. In order to convert this system into a well-posed system with

bounded input and output operators, the boundary condition of the system should

be converted to a homogeneous boundary condition and the input variable u should

appear in the dynamical equation of the system similar to system (4.15) . In order to

reformulate it to a well-posed abstract Cauchy problem, a new operator A is defined

by:

Ax = Ax

D(A) = D(A) \ ker(B)

= {x 2 H : x anddx

dzare a.c. ,

d2x

dz22 H

and D@x(0, t)

@z� V

2x(0, t) = 0, D

@x(l, t)

@z� V

2x(l, t) = 0}

(4.27)

Let us assume that A is the infinitesimal generator of a C0

-semigroup on H and

there exist a B such that for all u, Bu 2 D(A), and the following holds:

BBu = u, u 2 U (4.28)


In other words, B should satisfy the following conditions:

�DdB(0)

dz+

V

2B(0) = 1 (4.29a)

D@B(l)

@z=

V

2B(l) (4.29b)

Condition (4.29a) is equivalent to (4.28) and the condition (4.29b) is from the

assumption that Bu 2 D(A). B can be any arbitrary function that satisfies these

conditions.

By defining a new input u(t) = u(t) and a new state xe(t) =hu(t) x(t) � Bu(t)

i0, the problem can be reformulated on the extended state space

He = H � U as (Curtain and Zwart, 1995):

xe(t) = Aexe(t) + Beu(t)

ye = Cexe(4.30)

where

Ae =

"0 0

AB A

#, Be =

"I

�B

#, Ce = C

hB I

i(4.31)

The infinite dimensional system (4.30) is a well-posed system with bounded input and

output operators, which is an exact representation of the original infinite dimensional

system (4.18). Therefore, one can design the LQ controller for the well-posed system

and apply it to the original system without any approximation. Additionally, the

dynamical properties of the extended system are similar to the dynamical properties

of the original system. Hence, it is su�cient to study the dynamical properties of the

extended system.

Optimal Control Design

In this section, a linear-quadratic optimal controller is formulated for a system

described by (4.30). The state-feedback controller assumes that full state

measurement is available. The existence of the solution for the LQ control problem


requires that the linear system is exponentially stabilizable and detectable. These two

properties will be investigated and in the following sections it will be proven that the

infinite dimensional system (4.30) is a Riesz-Spectral system. In this section, we will

use the spectral properties of Riesz-Spectral systems to solve the LQ control problem.

The linear quadratic control problem on an inifinite-time horizon employs the cost

function:

J(x0

, u) =

Z 1

0

hy(s), y(s)i + hu(s), Ru(s)ids (4.32)

where u(s) and y(s) are the input and output trajectories, respectively, and R is a

self-adjoint, coercive operator in L(U). The output function y(·) is given by Equation

(4.30).

It is well-known that the solution of this optimal control problem can be obtained

by solving the following algebraic Riccati equation (ARE) (Curtain and Zwart, 1995):

hAex1

,⇧x2

i + h⇧x1

, Aex2

i + hCex1

, Cex2

i � hBe⇤⇧x1

, R�1Be⇤⇧x2

i = 0 (4.33)

When (Ae, Be) is exponentially stabilizable and (Ce, Ae) is exponentially detectable,

the algebraic Riccati equation (4.33) has a unique nonnegative self-adjoint solution

⇧ 2 L(H) and for any initial state x0

2 H the quadratic cost (4.32) is minimized by

the unique control uopt given by:

uopt(s; x0

) = �R�1Be⇤⇧xopt(s), xopt(s) = T�BeR�1Be⇤⇧

(s)x(0) (4.34)

In addition, the optimal cost is given by J(x0

, uopt) = hx0

,⇧x0

i.The ARE (4.33) is valid for any infinite dimensional system, but depending on the

characteristics of the infinite dimensional system di↵erent approaches are needed to

solve it. For Riesz spectral systems, the spectral properties of the system can be used

to solve the ARE (4.33). Let us denote by �n, the normalized eigenfunctions of Ae.

If we take z1

= �m and z2

= �n, then the Riccati equation (4.33) becomes:

hAe�n,⇧�mi + h⇧�n, Ae�mi + hCe�n, C

e�mi � hBe⇤⇧�n, R�1Be⇤⇧�mi = 0 (4.35)

If we assume that the solution ⇧ has the form ⇧z =P

n,m⇧nmhz, mi n where

⇧nm = h�n,⇧�mi, then Equation (4.35) becomes the system of infinite number of


coupled scalar equations:

(�n + �m)⇧nm + Cnm �1X

k=0

1X

l=0

⇧nk⇧lmBnm = 0 (4.36)

where Cnm = hCe�n, Ce�mi, and Bnm = hR�1Be n, B

e mi.Using the spectral properties of the Riesz Spectral systems, the operator Riccati

equation (4.33) is converted to a set of coupled algebraic equations that can be

solved by any numerical algorithm. Then the main step to solve the optimal control

problem for these systems is to calculate the spectrum of the operator Ae. This can

be performed by solving the eigenvalue problem for the operator Ae. In the following

section, the solution of the eigenvalue problem for system of Equation (4.30) will be

studied.

4.2.1 Eigenvalue Problem

This section is devoted to solving the eigenvalue problem of an infinite dimensional

system given by equation (4.30). The analytical solution of the eigenvalue problem for

a general form of the operator Ae is not possible. In this work, it is assumed that the

matrix M(z) of the operator A has a triangular form. This assumption is true if there

is only one way coupling between state variables. For example, in the case of systems

with two state variables x1

and x2

, the one way coupling means that x1

is a function of

x2

, but x2

is independent of x1

. For most chemical engineering processes the operator

Ae can be converted to a triangularized form using a transformation. Also, without

loss of generality, the eigenvalue problem is solved for a case with two state variables.

The extension of solution for more than two state variables is straightforward. The

eigenvalue problem of interest will be:

Ae� = �� (4.37)

where Ae has the following form:

Ae =

"0 0

AB A

#(4.38)


and

A =

"A

11

0

A12

A22

#=

"D

11

d2.dz2

+M11

(.) 0

M21

(.) D22

d2.dz2

+M22

(.)

#(4.39)

Eigenvalues and eigenfunctions of A11

and A22

A11

and A22

are both self-adjoint Sturm-Liouville operators. Assuming that the

operators A11

and A22

were Sturm-Liouville operators with constant coe�cients of

the form

@#

@t= D

@2#

@2z+M#

D@#(0, t)

@z=

V

2#

D@#(L, t)

@z= �V

2#(L, t)

(4.40)

their eigenvalues would be given by:

µ2

n = D!2

n +M (4.41)

where !n is the solution to the following equation:

tan(!l) =4D!V

4D2!2 � V 2

(4.42)

and the corresponding eigenfunctions are given by:

n = cos(!nz) +V

2D!n

sin(!nz) (4.43)

Recall that the linearization of nonlinear equations around the steady state profile

results in equations with spatially varying coe�cients. Therefore, calculation of

the spectrum of the operator Aii is a challenging issue. In this work, the method

of solving the heat equation for composite media (Friedman, 1976) is adopted to

solve the eigenvalue problem for a Sturm-Liouville operator with spatially varying

coe�cients. In this approach, the length of the reactor is divided into a finite number

of segments. It is assumed that within each segment, the values of coe�cients

are constant. Figure 4.1 illustrates the discretization method. The mathematical

formulation of this approach can be given as (de. Monte, 2002):


···

0

1

N···

i � 1

i

i + 1

N � 1

D��1(0, t)

�z= h1�1(0, t)

�i�1(zi, t) = �i(zi, t)✓��i�1

�z

◆

xi

=

✓��i

�z

◆

xi

D��N (L, t)

�z= �hN�N (L, t)

Figure 4.1: Approximation of the reactor as a composite media

@#i

@t= D

@2#i

@2z+ ki#i, #i = x

1

(z), z 2 [zi�1

, zi], i = 1, · · · , N (4.44)

The boundary condition at z = 0 is:

D@#

1

(0, t)

@z= h

1

#1

(0, t), h1

=v

2(4.45)

The continuity boundary conditions are:

#i�1

(zi, t) = #i(zi, t) (i = 2, 3, · · · , N) (4.46)✓@#i�1

@z

◆

xi

=

✓@#i

@z

◆

xi

(i = 2, 3, · · · , N) (4.47)

The boundary condition at x = L is:

D@#N(L, t)

@z= �hN#N(L, t), hN =

v

2(4.48)

The continuity conditions at the boundary imply that, at the interfaces of the

segments, the concentration and the flux are equal. This approach results in a set of N


PDEs coupled through boundaries. The PDEs should be solved for each segment with

unspecified boundary data at the interfaces. Then, the solutions can be determined

by applying the continuity conditions.

Equations (4.44)-(4.48) can be solved by the method of separation of variables. We

will have:

#i(z, t) = Zi(z)Gi(t), z 2 [zi, zi+1

] (4.49)

Substituting the above equation in the original PDE, we obtain:

D@2Zi

@2z+ kiZi = ��2iZi (4.50)

which can be written as:

D@2Zi

@2z+$2

iZi = 0, $2

i = ki + �2i ; (4.51)

Eigenfunctions associated with Equation (4.51) have the form:

Zi(z) = ai sin($ix) + bi cos($ix) (4.52)

ai and bi are integration constants and should be calculated by the boundary

conditions at each segment. Since the PDEs are coupled through boundary conditions

these coe�cients should be calculated simultaneously. Finally, Zi can be obtained as

(de. Monte, 2002):

Zi(z) = a1

�i($i) (sin($iz) + �i cos($iz)) , z 2 [zi, zi+1

] (4.53)

where

�1

= 1; �i($1

, · · · ,$i) = �i,i�1

�i�1,i�2

· · ·�2,1 (4.54)

and

�i,i�1

($1

, · · · ,$i) =(sin($i�1

zi) + �i�1

cos($i�1

zi))

(sin($izi) + �i cos($izi))(4.55a)

�M,M�1

($1

, · · · ,$M) =1

kM

(cos($M�1

zi) � �M�1

sin($M�1

zM))

(cos($MzM) � �M sin($MzM))(4.55b)


�1

= �h1

sin($1

z1

) � k1

$1

cos($1

z1

)

h1

cos($1

z1

) + k1

$1

sin($1

z1

)(4.56a)

�i =

"cos($izi) (sin($i�1

zi) + �i�1

cos($i�1

zi))

� sin($izi) (cos($i�1

zi) � �i�1

sin($i�1

zi))

#

"sin($izi) (sin($i�1

zi) + �i�1

cos($i�1

zi))

+ cos($izi) (cos($i�1

zi) � �i�1

sin($i�1

zi))

# (4.56b)

�M = �hM+1

sin($MzM+1

) + kM$M cos($MzM+1

)

hM+1

cos($MzM+1

) � kM$i sin($MzM+1

)(4.56c)

�M can be evaluated by both Equation (4.56b) for i = M and Equation (4.56c).

Therefore, one can compute � by comparing these equations. A solution can be

obtained by using numerical or graphical methods. There are an infinite number of

eigenvalues satisfying this condition and each has a corresponding eigenfunction of

the form given in Equation (4.53).

Eigenvalues and eigenfunctions of the operator A

The operator A is triangular and therefore, its eigenvalues consist of eigenvalues of A11

and A22

, �(A) = �(A11

) [ �(A22

). Let �n and �n be eigenvalues and eigenfunctions

of the operators A11

and µn and n be eigenvalues and eigenfunctions of the operator

A22

computed by the method described in section 4.2.1. Hence, �(A) is given by:

�2n+1

= �n, forn � 0 (4.57)

�2n = µn, forn � 1 (4.58)

The corresponding eigenvectors can be found by solving the eigenvalue problem for

the operator A. For example, if we define �2n+1

=

"&1,n

&2,n

#as eigenvectors associated

with �2n+1

we have

A�2n+1

= �2n+1

�2n+1

"A

11

0

A21

A22

# "&1,n

&2,n

#= �n

"&1,n

&2,n

#(4.59)


Equation (4.59) simplifies to

A11

&1,n = �n&1,n (4.60a)

A21

&1,n + A

22

&2,n = �n&2,n (4.60b)

Equation (4.60a) implies that &1,n is eigenfunction of A

11

or &1,n = �n. Rearranging

Equation (4.60b) results in

&2,n = (�n � A

22

)�1A21

�n (4.61)

Therefore

�2n+1

=

"�n

(�nI � A22

)�1A21

�n

#(4.62)

Following the same approach for �2n results in

�2n =

"0

n

#(4.63)

The corresponding biorthonormal eigenfunctions can be computed by a similar

approach and are:

2n+1

=

"�n

0

#(4.64)

2n =

"(µnI � A

11

)�1A21

n

n

#(4.65)

In the equations above, (µnI � A11

)�1 and (�nI � A22

)�1 are the resolvents of the

operators A11

and A22

, respectively, and can be calculated using Equation (4.9)

(µnI � A11

)�1A21

n =1X

m=0

1

µn � �mhA

21

n,�mi�m

(�nI � A22

)�1A21

�n =1X

m=0

1

�n � µm

hA21

�n, mi m

Eigenvalues and eigenfunctions of the operator Ae

Ae is a triangular operator with A and 0 as its diagonal elements, therefore �(Ae) =

�(A) [ {0}. The multiplicity of {0} is denoted by r0

and is equal to the number

Sec. 4.3 Stability Analysis 73

of manipulated variables. Following a similar approach to the previous section, the

corresponding eigenfunctions on the extended space are given by

�i0 =

"ei

�A�1(AB)

#=

"eiP1

m=0

1

�mhAB, mi�m

#, i = 1, . . . , r

0

(4.66)

and

�n =

"0

�n

#, n � 1 (4.67)

where �n is the eigenfunction of the operator A and ei, i = 1, . . . ,m is the orthonormal

basis function for U . Calculation of the associated biorthonormal eigenfunctions is

straightforward and are given by:

i0 =

"ei

0

#, i = 1, . . . , r

0

n =

"1

�n(AB)⇤ n

n

#

where �n and n are the eigenvalues and the corresponding eigenfunctions of A⇤. In

the equations above, �i0 and i0 are corresponding eigenfunctions and biorthonormal

eigenfunctions corresponding to eigenvalues equal to 0.

4.3 Stability Analysis

Theorem 4.3.1. Consider the family of operators {Ae(t)}t�0

given by Equations

(4.38)-(4.39). Then, {Ae(t)}t�0

is a stable family of infinitesimal generators of C0

-

semigroup on H.

Proof. Operator A has a lower triangular form and the diagonal elements of �Aare Sturm-Liouville operators. Therefore, A

11

, A22

are the infinitesimal generators

of C0

-semigroups T11

and T22

, respectively (Winkin et al. (2000), Delattre et al.

(2003)). As discussed in §4.2.1, eigenvalues of A consist of eigenvalues of A11

and A22

.

Therefore, the operator A has real, countable and distinct eigenvalues. Furthermore,

eigenfunctions of A and its adjoint are biorthogonal. Thus, the operator A is a Riesz


spectral operator(Delattre et al., 2003). By (Curtain and Zwart, 1995), Lemma 3.2.2,

the operator A is the infinitesimal generator of the C0

-semigroup T given by

T (t) =

"T11

(t) 0

T21

(t) T22

(t)

#(4.68)

where

T21

(t)x1

=

Z t

0

T22

(t � s)A21

T11

(s)x1

ds (4.69)

By the same approach, one can deduce that the operator Ae is a Riesz spectral

operator and generates a C0

- semigroup given by:

T e(t) =

"I 0

S(t) T (t)

#(4.70)

where S(t)x =R t

0

T (s)ABxds, and T (t) is the C0

-semigroup generated by A.

Theorem 4.3.2. Consider the state linear systemP

(A,B,�), where A is a Riesz-

Spectral operator on the Hilbert space Z with the representation

Az =1X

n=1

�n

rnX

j=1

hz, nji�nj (4.71)

{�n, n � 1} are eigenvalues of A with finite multiplicity rn, and {�nj, j =

1, · · · , rn, n � 1} and { nj, j = 1, · · · , rn, n � 1} are generalized eigenfunctions of A

and A⇤, respectively. B is a finite rank operator defined by

Bu =mX

i=1

biui, where bi 2 Z (4.72)

Necessary and su�cient conditions forP

(A,B,�) to be �-exponentially

stabilizable are that there exists an ✏ > 0 such that �+

��✏(A)comprises, at most, finitely

many eigenvalues and

rank

0

BB@

hb1

, n1i · · · hbm, n1i...

...

hb1

, nrni · · · hbm, nrni

1

CCA = rn (4.73)

for all n such that �n 2 �+

��✏(A).


Proof. Since �+

��✏(A) comprises at most finitely many eigenvalues, Z+

��✏ is finite

dimensional. Therefore, A satisfies the spectrum decomposition assumption (Curtain

and Zwart, 1995) and the corresponding spectral projection on the finite dimensional

subspace Z+

��✏ is given by

P��✏z =1

2⇡j

Z

��✏

(�I � A)�1zd� =X

�i2�+��✏

rnX

j=1

hz, nji�nj (4.74)

It can easily be calculated that

Z+

��✏ = span�n2�+

��✏

{�nj , j = 1, · · · , rn} (4.75)

Z��✏ = span

�n2��✏

{�nj , j = 1, · · · , rn} (4.76)

T��✏(t)z =

X

�n2��✏

e�nt

rnX

j=1

hz, nji�nj (4.77)

A+

��✏z =X

�n2�+��✏

�n

rnX

j=1

hz, nji�nj (4.78)

B+

��✏u =X

�n2�+��✏

rnX

j=1

hBu, nji�nj (4.79)

T��✏(t) satisfies the spectrum determined growth assumption and is �-exponentially

stable. Now we need to show that the finite dimensional systemP

(A+

��✏, B+

��✏,�)

is controllable, which is equivalent to proving that the reachability subspace ofP

(A+

��✏, B+

��✏,�) is dense in Z+

��✏. From Curtain and Zwart (1995, Definition

A.2.29), R is dense in Z+

��✏ if and only if R? = {0} where R? is the orthogonal

complement of R. R and R? are given by

R := {z 2 Z+

��✏| there exist ⌧ > 0 and u 2 L2

([0, ⌧ ];U) (4.80)

such that z =

Z ⌧

0

T+

��✏(⌧ � s)B+

��✏u(s)ds}

R? = {x 2 Z+

��✏|hx, yi = 0 for all y 2 R} (4.81)


In order to have R? = {0}, there should be no x 2 Z+

��✏ that is orthogonal to the

reachability subspace R. This means that for every x 2 Z+

��✏, there is a u such that

hx, R ⌧

0

T+

��✏(⌧ � s)B+

��✏u(s)dsi 6= 0. It can be shown that this condition is equivalent

to

X

�n

e�n(⌧�s)

rnX

j=1

hx,�njihBu, nji 6= 0 (4.82)

Thus for every �n 2 �+ and j = 1, · · · , rn there exists a u such thatrnPj=1

hx,�njihBu, nji 6= 0. This condition is equivalent to

BnUn 6= 0 (4.83)

Bn =

2

66664

hb1

, n1i hb2

, n1i · · · hbm, n1ihb

1

, n2i hb2

, n2i · · · hbm, n2i...

......

hb1

, nrni hb2


3

77775(4.84)

Un =

2

66664

hu1

, n1i hu1

, nrni · · · hu1

, nrnihu

2

, n1i hu2

, n2i · · · hu2

, nrni...

......

hum, n1i hum, n2i · · · hum, nrni

3

77775(4.85)

If rank(Bn) < rn, there exists a u 6= 0 such that BnUn = 0. Thus, the system is

uncontrollable if rank(Bn) < rn. If rank(Bn) = rn, the only solution for BnUn = 0 is

u = 0 and therefore R? = {0} and the system is controllable.

By duality,P

(A,�, C) is �-exponentially detectable, ifP

(A⇤, C⇤,�) is �-

exponentially stabilizable. Similarly, the systemP

(Ae,�, Ce) is �-exponentially

detectable.

Remark 4.3.1. Consider the infinite dimensional systemP

(Ae, Be, Ce) given by

Equation (4.30). As discussed in §4.2.1, the eigenvalues of Ae consist of eigenvalues

of A and 0 with finite multiplicity m. Since the diagonal entries of A are Sturm-

Liouville operators, the spectrum of A is finitely bounded (i.e., there exists a ! such

that all � 2 �(A) < !). Therefore, for any arbitraty �, �+

��✏(Ae) comprises finitely

Sec. 4.4 Case Study: A Fixed-Bed Reactor with Axial Dispersion 77

many eigenvalues and the first condition of Theorem (4.3.2) holds. The condition

(4.73) depends on the choice of B and should be verified for each case study.

4.4 Case Study: A Fixed-Bed Reactor with Axial

Dispersion

In this section, the proposed linear-quadratic optimal controller is applied to a tubular

catalytic cracking reactor. This process involves axial dispersion, convection and

reactions, and can be modelled by a set of parabolic equations. The controller is used

to regulate the distribution of the concentration of gasoline along the length of the

reactor by manipulating the inlet concentration.

4.4.1 Model Description

The model that is used to describe the catalytic cracking reactor is developed

by Froment (1990). This simplified model of the system considers three lumped

components: feed (gas oil), gasoline fraction and the remaining products (e.g.,

butanes, coke, dry gas). The reaction scheme is as following

Ak1�! B

k2�! C

Ak3�! C

(4.86)

where A represents gas oil, B gasoline, and C other products. The rate equations are

(Weekman, 1969):

rA = �(k1

+ k3

)y2A = �k0

y2A

rB = k1

y2A � k2

yB(4.87)

The schematic diagram of the reactor is shown in Figure 4.2. To model the

reactor, an isothermal process is considered. Therefore the dynamics of the system are

described by the following parabolic partial di↵erential equations (PDEs) representing


Ak1�! B

k2�! C

Ak3�! C

yAin , yBin yA(l, t), yB(l, t)

Figure 4.2: Schematic diagram of the catalytic cracking reactor

the component balances within the reactor:

@yA@t

= Da@2yA@z2

� v@yA@z

+ rA,

@yB@t

= Da@2yB@z2

� v@yB@z

+ rB

(4.88)

Initial and boundary conditions are:

Da@yA@z

|z=0

= v(yA|z=0

� yAin),

Da@yB@z

|z=0

= v(yB|z=0

� yBin),

@yA@z

|z=l = 0,

@yB@z

|z=l = 0,

yA(z, 0) = yA0(z),

yB(z, 0) = yB0(z)

(4.89)

In the equations above, yA, yB, Da, v, yAin , and yBin denote the weight fractions of

reactant A and B, the axial dispersion coe�cient, the superficial velocity, the inlet

weight fraction of component A, and the inlet weight fraction of component B,

respectively.

The corresponding steady-state equations of the PDE model defined by Equation

(4.88) are given by the following ordinary di↵erential equations:

Da@2yAss

@z2� v

@yAss

@z� k

0

y2Ass= 0,

Da@2yBss

@z2� v

@yBss

@z+ k

1

y2Ass� k

2

yBss = 0

(4.90)


Linearized model

Let us define the following state variables:

✓(t) =

"yA � yAss

yB � yBss

#(4.91)

and assuming the inlet weight fraction of A, yAin as the manipulated variable, the

new input is:

u(t) = v(yAin � yAin,ss) (4.92)

Then, the nonlinear system given in Equation (4.88) can be linearized around its

steady state profile to yield:

@

@t

"✓1

✓2

#=

"Da

@2

@z2� v @

@z� 2k

0

yAss 0

2k1

yAss Da@2

@z2� v @

@z� k

2

# "✓1

✓2

#(4.93)

with the initial and boundary conditions:

Da@

@z

"✓1

✓2

#

z=0

= v

"✓1

✓2

#

z=0

�"u

0

#,

Da@

@z

"✓1

✓2

#

z=l

=

"0

0

#,

✓(z, 0) = ✓0

=

"✓10(z)

✓20(z)

#=

"yA0 � yAss

yA0 � yAss

#(4.94)

The linearized model given by Equation (4.93) is a set of linear Sturm-

Liouville equations and can be converted into a di↵usion-reaction system with the

transformation given in Equation (4.19). The resulting equations are:

@

@t

"x1

x2

#=

"Da

@2

@z2� k

1

(z) 0

2k1

yAss(z) Da@2

@z2� k

2

# "x1

x2

#(4.95)

with the following boundary and initial conditions:

Da@

@z

"x1

x2

#

z=0

=v

2

"x1

x2

#

z=0

�"

u

0

#

Da@

@z

"x1

x2

#

z=l

= �v

2

"x1

x2

#

z=l

x(z, 0) = x0

(4.96)

where x0

= e�v2D z✓

0

, k1

(z) = � v2

4Da� 2k

0

yAss(z) , and k2

= � v2

4Da� k

2

.


Infinite-Dimensional representation

Equations (4.95)-(4.96) can be formulated as an abstract boundary control problem

on the Hilbert space H := L2(0, l) ⇥ L2(0, l) given by (4.22),

The operator A is given as follows:

A =

"Da

d2

dz2� k

1

(z) 0

2k1

yAss(z) Dad2

dz2� k

2

#:=

"A

11

0

A21

A22

#(4.97)

with the domain:


dzare a.c. ,

d2x

dz22 H and Da

@x1

(l, t)

@z+

v

2x1

(l, t) = 0,

(4.98)

Da@x

2

(0, t)

@z� v

2x2

(0, t) = 0, Da@x

2

(l, t)

@z+

v

2x2

(l, t) = 0} (4.99)

where a.c. means that x is absolutely continuous. The boundary operatorB : H ! R

is given as:

Bx(z) =h

�Daddz

+ v2

0i "

x1

(0)

x2

(0)

#(4.100)

with the domain:

D(B) = {x 2 H : x is a.c. ,dx

dz2 H} (4.101)

and the output operator, C, is defined by the available measurement.

This abstract boundary control problem is transformed to a well-posed system as

discussed in section §4.2. The new operator A is defined by:

Ax = Ax

D(A) = D(A) \ ker(B)

= {x 2 H : x anddx

dzare a.c. ,

d2x

dz22 H

and Da@x(0, t)

@z+

v

2x(0, t) = 0, Da

@x(l, t)

@z+

v

2x(l, t) = 0}

(4.102)

The function B in equation (4.28) is an arbitrary function satisfying the conditions

(4.29a) and (4.29b). Here, it is assumed that B is a vector of two second order

polynomials and the coe�cients are found using the conditions (4.29a) and (4.29b)


as following

B =

"B

1

B2

#(4.103)

B1

=�2

4Dal + vl2z2 +

2

v(4.104)

B2

= � 1

4Dal + vl2z2 +

1

4Da + vlz +

2Da

4Dav + v2l(4.105)

Finally, the new well posed system can be formulated using the procedure given in

§4.2, and the optimal control problem can be solved as discussed in section §4.2

4.4.2 Simulation Study

In this section the closed loop performance of the proposed approach is demonstrated.

Values of the model parameters are given in Table 4.1 (Weekman, 1969).

The LQ-controller discussed in the previous section was studied via a simulation

that used a nonlinear model of the reactor given in Equations (4.87)-(4.89). The

performance of the proposed controller was compared to a finite dimensional LQ

controller. The finite dimensional LQ controller was computed using the model

derived by converting the PDEs to ODEs by applying central finite di↵erence

approach.

Table 4.1: Model ParametersParameter Value Unit

k0

22.9 (hr⇥weight fraction)�1

k1


k2

1.7 hr�1

Da 0.5 m2hr�1

v 2 m⇥hr�1

yAin 0.7 weight fraction

yBin 0 weight fraction

The control objective is to drive the trajectory of yB to the desired steady state

profile. Using the nominal operating conditions, and the model given in Equations


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

z (m)

yA

ss

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.16

0.18

0.2

0.22

0.24

0.26

0.28

0.3

0.32

yB

ss

Figure 4.3: Steady state profile of yA and yB

(4.87)-(4.89), the steady-state profiles of yA and yB were computed and are shown in

Figure 4.3. Then, the nonlinear model was linearized around the stationary states and

transformed to the self-adjoint form of Equations (4.95)-(4.96). Spectra of operators

A11

and A22

were calculated using the algorithm discussed in §4.2.1. In order to

compute the spectrum of A11

, it was assumed that the length of the reactor is divided

into 50 equally-spaced sections and the coe�cient of the reaction term is constant in

each section. First five eigenvalues of the operator A11

are:

� = {�2.39 ⇥ 10�5,�1.34 ⇥ 10�4,�4.46 ⇥ 10�4,�1.12 ⇥ 10�3,�2.35 ⇥ 10�3}

The spectrum of A22

can be computed using Equations (4.41)-(4.43). First five

eigenvalues of A22

are:

� = {�2.04 ⇥ 10�6,�1.096 ⇥ 10�5,�5.68 ⇥ 10�5,�2.08 ⇥ 10�4,�5.78 ⇥ 10�4}

Finally, the spectrum of Ae was computed using Equations (4.66)-(4.67). The

results are shown in Figures 4.4-4.5. Once the eigenvalues and eigenfunctions of


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1!2.5

!2

!1.5

!1

!0.5

0

0.5

1

1.5

2

2.5

3

z (m)

�n(2

)

�0(2) ⇥ 10�2 �1(2) �3(2)

Figure 4.4: Second element of �n given by Equation (4.67) and Equations (4.62)-(4.63)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

!3

!2

!1

0

1

2

3

z (m)

�n(3

)

�n(3) ⇥ 10�3 �1(3) �2(3) �3(3) �4(3)

Figure 4.5: Third element of �n given by Equation (4.67) and Equations (4.62)-(4.63)


the operator Ae are calculated, the LQ-feedback controller can be computed using

Equation (4.36). Note that since ⇧ is a self-adjoint operator, h�n,⇧�mi = h�m,⇧�ni,therefore ⇧nm = ⇧mn. As a result, Equation (4.36) gives n(n+1)

2

coupled algebraic

equations that should be solved simultaneously where n is the number of modes that

are used to formulate the controller. Since there are two orders of magnitude di↵erence

between the first and fifth eigenvalues of the operator Ae, the e↵ect of higher order

eigenvalues on the dynamic of the system is considered to be negligible; therefore,

in this work, the first five modes were used for numerical simulation. The computed

LQ controller was applied to the nonlinear model of the reactor. Simulation of the

nonlinear system was performed using COMSOL R�. The closed loop trajectory of yB

is shown in Figure 4.6. It should be mentioned the simulation based on first 21 modes

resulted in identical results.

01000

20003000

40005000

0

0.2

0.4

0.6

0.8

10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

t (s)z (m)

yB

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Figure 4.6: Closed loop trajectory of yB

In order to investigate the performance of the controller, a LQ controller based on

a finite di↵erence method was designed. A very fine discretization (n=50) is used

to develop the controller. Tuning of the LQ controller based on finite di↵erence was


challenging for this case study. Small values of input weight resulted in infeasible

values for manipulated variable. Since the manipulated variable is the inlet weight

fraction of component A, its value should be between 0 and 1. In Figure 4.7 the

trajectory of the manipulated variable is shown for three di↵erent cases. As it is

illustrated for R = 5 ⇥ 105 the value of yAin is greater than 1. This problem can be

solved by increasing the weight on the input. As it is shown in Figure 4.7, the optimal

input trajectory is feasible for R = 1 ⇥ 106.

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000.65

0.7

0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

t (s)

yA

in

Infinite dimensional

Finite difference, R=5! 105


Figure 4.7: Closed loop trajectory of the input variable (Inlet Concentration)

Closed loop trajectory of yB for R = 1 ⇥ 106 is shown in Figure 4.8. To compare

the performance of two approaches, the l2-norm of the error between the steady-

state and closed loop profiles have been calculated. The results are illustrated in

Figure 4.9. It can be observed that the l2-norm of the error for infinite dimensional

case is smaller than either finite dimensional cases. By increasing the weight on the

input for finite dimensional case, the performance deteriorates. The l2-norm of the

errors for finite and infinite dimensional controllers are given in Table 4.2. It can

be observed that this value for infinite dimensional case is 16% better than the best


finite dimensional controller which is physically meaningful (i.e., results in values of

manipulated variable between 0 and 1).

01000

20003000

40005000

0

0.5

10

0.1

0.2

0.3

0.4

0.5

0.6

t (s)z (m)

yB

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Figure 4.8: Closed loop trajectory of yB for finite dimensional controller

Table 4.2: Comparison of the l2-norm

Controller l2-norm

Infinite dimensional controller 1771

Finite dimensional controller with R = 5 ⇥ 105 1903

Finite dimensional controller with R = 1 ⇥ 106 2064

Designing of the infinite dimensional controller needs more e↵ort than the finite

dimensional controller, as it requires solution of eigenvalue problem given in §4.2.1.On the other hand the infinite dimensional controller results in better performance

than finite dimensional one. One might claim that by increasing the number of

discretization points in finite di↵erence algorithm, the finite dimensional controller

may result in better performance. Since the infinite dimensional controller involves

matrix algebra, increasing the size of the matrices results in increase in online

Sec. 4.5 Summary 87

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

t (s)

no

rm o

f err

or

Infinite dimensional



Figure 4.9: Trajectory of tracking error

computation e↵ort: however, the infinite dimensional controller does not involve

matrix algebra. Thus by increasing the number of modes the computation e↵ort

will not increase significantly. It should also be mentioned that, unlike the finite

dimensional controller, the infinite dimensional controller results in feasible optimal

trajectories for all values of R. Di�culty in tuning the controller, computational

e↵ort and poor performance of the finite dimensional controller justifies implementing

the infinite dimensional controller. It should be noted that the infinite dimensional

controller requires more e↵ort for design but leads to a better performance.

4.5 Summary

In this chapter, optimal boundary control of infinite dimensional systems described

by a set of parabolic PDEs with spatially varying coe�cients was studied. Using

an exact transformation, the boundary control problem was transformed to a well-

posed abstract system. It was shown that the resulting system is a Riesz Spectral

Sec. 4.5 Summary 88

system. Stabilizability of the system was investigated using the spectral properties

of the system. In order to compute the spectra of the system, a method similar to

solution of the heat equation for composite media was used to compute the spectrum

of the system. Moreover, by using the spectral properties of the system, the operator

Riccati equation was converted to a set of algebraic equations that can be solved

numerically.

A tubular reactor with axial dispersion was considered as the case study. The

reactor can be modelled by a set of nonlinear parabolic partial di↵erential equations.

Linearization around the steady state profile of the system resulted in a set of linear

PDEs with spatially varying coe�cients. Then, an LQ controller was formulated

to regulate the output variable to the steady state profile. The performance of

the proposed controller was compared to an LQ controller based on finite di↵erence

approximation of the PDEs. Simulation results showed that the infinite dimensional

controller leads to better performance in terms of the l2-norm of the errors.

5LQ Control of Coupled Parabolic

PDE-ODE systems

In this chapter optimal control of a di↵usion-convection-reaction system with

catalyst deactivation modelled by a set of ODEs is studied. The system can

be described by a set of coupled parabolic PDEs and ODEs. This chapter is

the extension of the optimal controller discussed in Chapter 4. Most industrial

fixed-bed reactors, as well as fluidized-bed reactors, fall into this class of

distributed parameter systems and therefore, the contributions of this chapter

have significant industrial importance. The focus in this chapter is on catalytic

reactors, but the developed theory is applicable to any distributed parameter

system that can be described by the studied form of coupled PDE-ODE systems.

89

90

This chapter addresses infinite dimensional LQ control of the systems that are

described by a set of coupled parabolic PDEs and ODEs. These kind of systems arise

in modeling of chemical and bio-chemical processes, where part of the process is a

distributed parameter system and part of it is a lumped parameter system. Examples

of these systems include a tubular reactor followed by a well mixed reactor or a

catalytic reactor with catalyst deactivation, where the rate of the catalyst deactivation

is modelled by a set of ODEs. Two types of coupling can exist between PDEs and

ODEs. One type of coupling is through the boundary condition, where the boundary

condition of the distributed portion is a function of the state variables of the lumped

parameter system. A tubular and a well mixed reactor in series is a simple example for

this coupling. These kind of systems are called cascaded PDE-ODE systems and their

control was the subject of few recent studies (Krstic, 2009; Susto and Krstic, 2010).

The second type of coupling takes place in the domain of the PDE, which means the

parameters of the distributed part (e.g., the coe�cients) are functions of the states

of the lumped parameter system. Examples of this kind of coupling include catalytic

reactor with catalyst deactivation, where the deactivation kinetics is described by

a set of ODEs, or a heat exchanger with a time varying heat transfer coe�cient.

Most biochemical processes are also modelled by a set of coupled PDE-ODE with

in-domain coupling (e.g., in-situ bioremediation). To the best of authors knowledge,

there is no published work on the infinite dimensional optimal control of coupled

parabolic PDEs-ODEs with in-domain coupling and this work is the first step on the

study of infinite dimensional optimal controller for these systems.

In this chapter, the distributed parameter part of the system is assumed to be

similar to the infinite dimensional system studied in Chapter 4 and it is assumed that

the parameters of the reactive term are modelled by a set of ODEs which represent

the deactivation kinetics. Similar to Chapter 4, we are interested in the boundary

control of the system.

By introducing a transformation, the coupled PDE-ODE system is converted to

decoupled subsystems. Conditions on the existence of this transformation is studied.

Sec. 5.1 Coupled PDE-ODE Systems: Mathematical Description 91

Then, by using the transformation introduced in Chapter 4, the boundary control

system is converted to a well-posed infinite dimensional system with bounded input

and output operators. The method of computing the spectrum of the operator

introduced in Chapter 4 is extended to address the PDE-ODE system. It is shown that

the resulting infinite dimensional system is a Riesz spectral system and its stability

condition is investigated. Finally, the state feedback LQ controller is formulated for

this system and its performance is analyzed using numerical simulations.

This chapter is an important step for optimal control of the most general form of

the distributed parameter systems. The main contributions of this chapter can be

summarized as

• Formulation and solution of an optimal boundary control problem for coupled

parabolic PDE-ODE systems with in-domain coupling;

• Stability analysis of coupled parabolic PDE-ODE system with in-domain

coupling.

5.1 Coupled PDE-ODE Systems: Mathematical

Description

The problem of interest in this chapter is a system that consists of a set of parabolic

PDEs coupled with a set of ODEs. Since most chemical processes are nonlinear in

nature, we will start with nonlinear model of the system and will then linearize the

model around the steady state profile of the system. Following a few transformations

that will be introduced in this section, the linearized coupled PDE-ODE system will

be formulated as a well-posed infinite dimensional system with bounded input and

output operators, which is suitable for controller formulation.

For the rest of this chapter, it is assumed that the PDE part of the system is similar

to what we have considered in Chapter 4. It is also assumed that the parameters of the

kinetic terms in the parabolic equations are variable and modelled by a set of ODEs.

The mathematical representation of such a system can be given as the following set


of quasi-linear parabolic PDEs and nonlinear ODEs:

@X

@t= D

@2X

@z2� V

@X

@z+R(k,X)

dk

dt= f(k)

(5.1)

With the following initial and boundary conditions

D@X

@z|z=0

= V(Xz=0

� Xin)

@X

@z|z=l = 0

X(z, 0) = X0

k(0) = k0

(5.2)

where X(., t) =hX

1

(., t) · · · Xn(., t)iT

2 H := L2(0, l)n denotes the vector of

state variables of the distributed parameter portion, k =hk1

(t) · · · km(t)iT

2K := Rm denotes the vector of state variables for the lumped parameter portion of

the model. z 2 [0, l] 2 R and t 2 [0,1) denote position and time, respectively. D and

V are matrices of appropriate sizes. R is a Lipschitz continuous nonlinear operator

from H � K into H. f is a matrix of appropriate size whose entries are functions

defined in R. In a catalytic reactor, R represents the reaction term and f represents

the deactivation kinetics.

The nonlinear system (5.1)-(5.2) can be linearized around the steady state profile

and the resulting linear system is:

@X 0

@t= D

@2X 0

@z2� V

@X 0

@z+N

1

(z)X 0 +N2

(z)k0 (5.3a)

dk0

dt=

df(k)

dk|ssk0 (5.3b)


the boundary and initial conditions are:

D@X 0

@z|z=0

= V(X 0z=0

� X 0in)

@X 0

@z|z=l = 0

X 0(z, 0) = X 00

k0(0) = k00

(5.4)

where 0 indicates that the variables are in deviation form and N1

(z) = @R(k,X)

@X|ss and

N2

(z) = @R(k,X)

@k|ss.

The equation (5.3a) is a di↵usion-convection-reaction system. To simplify the

solution of the eigenvalue problem, it can be converted to a di↵usion-reaction system

by the following transformation

✓ = TX 0 = exp(�D�1V

2z)X 0 (5.5)

The resulting system is:

@✓

@t= D

@2✓

@z2+M

1

(z)✓ +M2

(z)k0 (5.6a)

dk0

dt= M

3

k0 (5.6b)

where M1

(z) = (TN1

(z)T�1 � VD�1V4

), M2

(z) = TN2

(z) and M3

= df(k)dk

|ss. The

boundary and initial conditions are:

D@✓

@z|z=0

=V

2✓z=0

� VX 0in

D@✓

@z|z=l = �V

2✓z=l

✓(z, 0) = ✓0

= T�1X 00

k0(0) = k00

(5.7)

Assuming that u = VX 0in, and by defining a new state vector

x =

"xd

xl

#=

"✓

k0

#(5.8)


the system (5.6)-(5.7) can be formulated as an abstract boundary control problem on

the space H = H � K given by

8>>>>>><

>>>>>>:

dx(t)dt

= Ax(t)

x(0) = x0

Bx(t) = u(t)

y(t) = Cx(t)

(5.9)

where A is a linear operator defined on the domain

D(A) = {x 2 H : xd anddxd

dzare a.c.,

dx2

d

dz22 H,D

dxd

dz|z=l = �V

2xd|x=l} (5.10)

and is given by

A =

"D 0

0 0

#d2.

dz2+

"M

1

M2

0 M3

#.I :=

"A

11

A12

0 A22

#(5.11)

The boundary operator B : H ! U := Rn is given by

Bx(.) =h

�D @@z

+ V2

0i "

xd

xl

#

z=0

(5.12)

Since M2

in Equation (5.11) is generally non-zero, xd and xl are coupled. By

introducing the following transformation, the system can be transformed to decoupled

subsystems:

⇤ =

"I J

0 I

#2 L(H) & x = ⇤x (5.13)

The operator A will be transformed to

A = ⇤A⇤�1 =

"A

11

�A11

J + A12

+ JA22

0 A22

#(5.14)

with D(A) = D(A). Therefore, if there exists a J that satisfies the following equation,

operator A will be decoupled.

�A11

J + A12

+ JA22

= 0 (5.15)


Remark 5.1.1. The Equation (5.15) is a Sylvester equation and admits a unique

solution if and only if �(�A11

) \ �(A22

) = ;. (Laub, 2005)

The solution is given by

J =

Z 1

0

T11

(t)A12

T12

(t)dt (5.16)

where T11

and T22

are C0

-semigroups generated by �A11

and A22

respectively

(Emirsajlow, 1999).

The resulting decoupled system is

8>>>>>><

>>>>>>:

dx(t)dt

= Ax(t)

x(0) = x0

Bx(t) = u(t)

y(t) = Cx(t)

(5.17)

where B = B⇤�1 and C = C⇤�1

The boundary control problem (5.17) is in the form of a standard abstract boundary

control problem and following a similar approach to §4.2, the abstract boundary

control problem (5.17) can be converted to a well-posed infinite dimensional system

with bounded input and output operators. The procedure is given below.

Define a new operator A by

Ax = Ax

D(A) = D(A) \ ker (B)

= {x 2 H : x anddx

dzare a.c.,

d2x

dz22 H

and Ddxd

dz|z=0

=V

2xd|z=0

,Ddxd

dz|z=l = �V

2xd|z=l}

(5.18)

If A generates a C0

-semigroup on H and there exist a B such that the following

conditions hold:

Bu 2 D(A) (5.19a)

BBu = u, u 2 U (5.19b)


the system (5.17) can be transformed into the following infinite dimensional system

with bounded input operator on the state space He = H � U

xe(t) = Aexe(t) + Beu(t)

ye(t) = Cexe(t)(5.20)

where

Ae =

"0 0

AB A

#, Be =

"I

�B

#, Ce = C

hB I

i(5.21)

and u(t) = u(t) and xe(t) =hu(t) x(t) � Bu(t)

iTare the new input and state

variables.

Remark 5.1.2. Consider B = ⇤�1B, then the conditions (5.19a)-(5.19b) become

Bu 2 D(A) and BBu = u, respectively. One can assume that B =hBd Bl

iT,

then the following conditions are equivalent to (5.19a)-(5.19b).

DdBd(0)

dz� V

2Bd(0) = I

DdBd(l)

dz+

V

2Bd(l) = 0

Bl 2 K

(5.22)

Bd can be any function that satisfies the above conditions. For simplicity one can

assume that Bd is a matrix of polynomials. Bl is any arbitrary matrix in K. Finally

B can be calculated by

B = ⇤B =

"Bd + JBl

Bl

#(5.23)

The infinite dimensional system (5.20) is in the form of a standard infinite

dimensional system and now we are in the position to proceed with stability analysis

and controller formulation for this system. It should be mentioned that, since all of the

transformations introduced in this chapter are exact transformations and there was

no approximation involved, all of the dynamical properties of the original linearized

Sec. 5.2 Coupled PDE-ODE Operator: Eigenvalue Problem 97

system are preserved. Hence, we can perform stability analysis and controller

formulation on the transformed system (5.20), and then apply the designed controller

to the original system. In order to perform the stability analysis, we need to solve

the eignenvalue problem for the system (5.20).

5.2 Coupled PDE-ODE Operator: Eigenvalue

Problem

In this section the solution of the eigenvalue problem that was introduced in Chapter

4 is extended to the case of coupled PDE-ODE systems. As discussed in Chapter

4, there is no general algorithm for analytical solution of eigenvalue problem for a

general form of parabolic operator. Therefore, in this section we will consider the

following assumptions:

1. N1

in (5.3a) is lower triangular, which leads to lower triangular form of A11

.

For most of chemical engineering processes, one can use a transformation to

triangularize the system.

2. The number of state variables in (5.3a) is two. Extension to more than two

variables is straightforward.

3. Without loss of generality, N3

is diagonal. Note that, N3

is a matrix and is

always diagonalizable, if its eigenvalues are simple.

Then the eigenvalue problem of interest will be:

Ae� = �� (5.24)

where Ae has the following form

Ae =

"0 0

AB A

#(5.25)


and

A =

"A

11

0

0 A22

#

A11

:=

"F11

0

F21

F22

#=

"D

11

d2

dz2+M

11

(z) 0

M21

(z) D22

d2

dz2+M

22

(z)

#

A22

=

"↵11

0

0 ↵22

#(5.26)

Eigenvalues and Eigenfunctions of AThe operator A is a block diagonal operator, therefore �(A) = �(A

11

) [ �(A22

).

Since A11

is a lower triangular operator �(A11

) = �(F11

) [ �(F22

). Then, �(A) =

�(F11

) [ �(F22

) [ �(A22

)

The operator A11

is similar to operator A in Chapter 4 and its spectrum can be

calculated by the same approach. Let �n and �n be eigenvalues and eigenfunctions

of the operator F11

and µn and n be eigenvalues and eigenfunctions of the operator

F22

. Then according to §4.2.1, the eigenvalues of the operator A11

are

�(A11

) = �(F11

) [ �(F2

) = {�n, µn} n = 1, · · · ,1 (5.27)

and the associated eigenvectors are:

("�n

(�nI � A22

)�1A21

�n

#,

"0

n

#)n = 1, · · · ,1 (5.28)

The corresponding bi-orthonormal eigenfunctions can be computed by solving the

eigenvalue problem for A⇤11

and are given by:

("�n

0

#,

"(µnI � A

11

)�1A21

n

n

#)n = 1, · · · ,1 (5.29)

A22

is a diagonal matrix and its eigenvalues are {↵11

,↵22

} with the associated

eigenvectors {[1, 0]T , [0, 1]T}. Finally:

�(A) = {�n, µn,↵11

,↵22

} n = 1, · · · ,1 (5.30)


and the associated eigenfunctions are8>>>><

>>>>:

2

66664

�n

(�nI � A22

)�1A21

�n

0

0

3

77775,

2

66664

0

n

0

0

3

77775,

2

66664

0

0

1

0

3

77775,

2

66664

0

0

0

1

3

77775

9>>>>=

>>>>;

n = 1, · · · ,1 (5.31)

The corresponding bi-orthonormal eigenfunctions are

8>>>><

>>>>:

2

66664

�n

0

0

0

3

77775,

2

66664

(µnI � A11

)�1A21

n

n

0

0

3

77775,

2

66664

0

0

1

0

3

77775,

2

66664

0

0

0

1

3

77775

9>>>>=

>>>>;

n = 1, · · · ,1 (5.32)

Eigenvalues and Eigenfunctions of Ae

Assume that the operator A has eigenvalues {�k, k � 1} and biorthonormal pair

{(�k, k), k � 1}. The operator Ae is similar to operator Ae in Chapter 4, and its

spectrum is calculated by the same approach. �(Ae) = �(A) [ {0} and �0

= 0 has a

multiplicity ofm, wherem is the number of manipulated variables. The corresponding

eigenfunctions for �0

are

�i0

=

"ei

�A�1(AB)

#=

"eiP1

k=0

1

�khAB, ki�k

#, i = 1, · · · ,m (5.33)

where ei, i = 1, · · · ,m is the orthonormal basis for U = Rn.

The corresponding bi-orthonormal eigenfunction of Ae is

i0

=

"ei

0

#, i = 1, · · · ,m (5.34)

For � 2 �(A), the associated bi-orthonormal pair are

�n =

"0

�n

#& n =

"1

�n(AB)⇤ n

n

#(5.35)

The procedure for solution of eigenvalue problem for operator Ae can be

summarized as:

Sec. 5.3 Stability Analysis of PDE-ODE Systems 100

• Compute the spectra of the operators F11

and F22

using the method proposed

in Chapter 4

• Given the spectra of F11

and F22

, compute the spectrum of A11

using Equations

(5.27)-(5.29)

• Compute spectrum of A22

• Given the spectra of A11

and A22

, compute the spectrum of A using Equations

(5.30)-(5.32)

• Compute the spectrum of Ae using Equations (5.33)-(5.35)

Now we have all the required information for stability analysis of the system, which

is the topic of the following section.

5.3 Stability Analysis of PDE-ODE Systems

Theorem 5.3.1. Consider operator A given by Equation (5.26). Then A is an

infinitesimal generator of C0

-semigroup on H.

Proof. Operators �F11

and �F22

, the diagonal entries of operator �A11

, are Sturm-

Liouville operators. Therefore, �F11

and �F22

are infinitesimal generators of C0

-

semigroups T11

and T22

, respectively. They both have real, countable and distinct

eigenvalues.

Operator A11

has a lower triangular form and its eigenvalues consist of eigenvalues

of F11

and F22

. Therefore, the operator A11

also has real, countable and distinct

eigenvalues. Furthermore, eigenfunctions of A11

and its adjoint are bi-orthonormal.

Thus, the operator A11

is a Riesz spectral operator (Delattre et al., 2003). By Curtain

and Zwart (1995), Lemma 3.2.2, the operator A11

is the generator of the C0

-semigroup

T11

given by

T11

(t) =

"T11

(t) 0

T21

(t) T22

(t)

#(5.36)

Sec. 5.3 Stability Analysis of PDE-ODE Systems 101

where

T21

(t)x1

=

Z t

0

T22

(t � s)F21

T11

(s)x1

ds (5.37)

Operator A22

, is a diagonal finite dimensional operator and therefore is the

generator of the semigroup T22

= exp(�A22

t). Thus, the operator A is the

infinitesimal generator of the following C0

-semigroup

T (t) =

"T11

(t) 0

0 T22

(t)

#(5.38)

Theorem 5.3.2. Consider operator Ae given by Equation (5.20). Then Ae is an

infinitesimal generator of C0

-semigroup on He.

Proof. Eigenvalues of operator Ae consist of eigenvalues of A and 0 with finite

multiplicity m. It is shown in Theorem 5.3.1 that eigenvalues of A are real,

countable and distinct. Therefore, Ae has real, countable eigenvalues with finite

multiplicity. Moreover, generalized eigenfunctions of Ae and its adjoint are bi-

orthonormal. Therefore, generalized egienfunctions of Ae form a Riesz basis for He.

Furthermore, Ae is a Riesz spectral operator and is the generator of a C0

-semigroup

given by

T e(t) =

"I 0

S(t) T (t)

#(5.39)

where S(t)x =R t

0

T (s)ABxds, and T (t) is the C0

-semigroup generated by A.

Theorem 5.3.3. Consider the state linear systemP

(Ae, Be, Ce) given by Equation

(5.20)

A necessary and su�cient condition forP

(Ae, Be,�) to be �-exponentially

stabilizable is

rank

0

BB@

hb1

, n1i · · · hbm, n1i...

...

hb1


1

CCA = rn (5.40)

Sec. 5.4 LQ Control of PDE-ODE Systems 102

for all n such that �n 2 �+

� (A).

Proof. Ae is a Riesz-Spectral operator and its eigenvalues consist of eigenvalues of

A and 0 with finite multiplicity m. Based on Theorem (4.3.2), the necessary and

su�cient conditions forP

(Ae, Be,�) to be ��exponentially stable are that there

exist an ✏ > 0 such that �+

��✏(Ae) comprises, at most, finitely many eigenvalues and

the rank condition (5.40) holds.

Diagonal entries of A are Sturm-Liouville operators, the spectrum of A is finitely

bounded (i.e., there exists a ! such that all � 2 �(A) < !). Therefore, for any

arbitrary � and ✏, �+

��✏(Ae) comprises finitely many eigenvalues and the first condition

holds. Therefore, the necessary and su�cient condition for ��exponential stability

of �(Ae, Be,�) reduces to (5.40). Validity of this condition depends on the choice of

B and should be verified for each case study.

5.4 LQ Control of PDE-ODE Systems

The extended state space system (5.20) is similar to the extended system (4.30),

therefore formulation of the LQ controller is similar to §4.2. The Algebraic Riccati

Equation (4.33) is independent of the type of operators, but its solution depends on

the properties of the infinite dimensional system. In the previous section we discussed

that the infinite dimensional system (5.20) is a Riesz Spectral system and therefore

the solution of optimal control problem for this system can also be found by solving

the set of algebraic equations given in (4.36).

In the following section, a case study involving a catalytic reactor with catalyst

deactivation will be investigated and the performance of the optimal controller will

be explored.

Sec. 5.5 Case study: Catalytic Cracking Reactor with Catalyst Deactivation 103

5.5 Case study: Catalytic Cracking Reactor with

Catalyst Deactivation

In this section, the proposed approach is applied to the catalytic cracking reactor

discussed in Chapter 4 with the assumption that the catalyst deactivates with time.

The control objective is to regulate the trajectory of CB, gasoline, at the steady

state profile using the inlet concentration of A as the manipulated variable. The

system will be represented as an infinite dimensional system using the procedures

discussed in previous sections. The transformations required to decouple the PDE-

ODE system and to convert the system to a well-posed infinite dimensional system will

be computed in detail. Finally, numerical simulations will be performed to analyze

the closed loop performance of the controller.

Model description

The model that is used is similar to the model of the catalytic cracking reactor in

Chapter 4. The only di↵erence is that the deactivation of the catalyst should also be

considered. The reaction scheme is given by Equations

Ak1�! B

k2�! C

Ak3�! C

(5.41)

with the kinetic equations given by

rA = �(k1

+ k3

)y2A = �k0

y2A

rB = k1

y2A � k2

yB(5.42)

It is assumed that the catalyst deactivation will only a↵ect the pre-exponential

factor of the main reaction, and k1

will be modelled by

dk1

dt= ↵k

1

+ �, k1

(0) = k10 (5.43)

The above equation for the rate of deactivation of the catalyst, is equivalent to the

exponential decay assumption, which is a common assumption for modelling catalyst


deactivation. It is in agreement with the observation that the catalyst deactivation

consists of three phases: rapid initial deactivation, slow deactivation and stabilization

(Kallinikos et al., 2008).

The model of the reactor will consist of Equations (4.88)-(4.89) and (5.43). The

linearization of the model can be performed similar to Chapter4. By defining the new

state and input variables

✓(t) =

2

64yA � yAss

yB � yBss

k1

� k1ss

3

75 , u(t) = v(yAin � yAin,ss) (5.44)

the linearized model will have the form of

@

@t

2

64✓1

✓2

✓3

3

75 =

2

64Da

@2

@z2� v @

@z� 2(k

1ss + k3

)yAss 0 �y2Ass

2k1ssyAss Da

@2

@z2� v @

@z� k

2

y2Ass

0 0 ↵

3

75

2

64✓1

✓2

✓3

3

75

(5.45)

with the initial and boundary conditions:

Da@

@z

"✓1

✓2

#

z=0

= v

"✓1

✓2

#

z=0

�"

u

0

#,

Da@

@z

"✓1

✓2

#

z=l

=

"0

0

#,

✓(z, 0) = ✓0

=

2

64✓10(z)

✓20(z)

✓30

3

75 =

2

64yA0 � yAss

yA0 � yAss

k10 � k

1ss

3

75

(5.46)

The set of equations (5.45) can be converted to a di↵usion-reaction system by using

the transformation given by Equation (5.5). The resulting system is

@

@t

2

64x1

x2

x3

3

75 =

2

64Da

@2

@z2� k

1

(z) 0 �y2Ass

2k1ssyAss(z) Da

@2

@z2� k

2

y2Ass

0 0 ↵

3

75

2

64x1

x2

x3

3

75 (5.47)


with the following initial and boundary conditions

Da@

@z

"x1

x2

#

z=0

=v

2

"x1

x2

#

z=0

�"

u

0

#

Da@

@z

"x1

x2

#

z=l

= �v

2

"x1

x2

#

z=l

x(z, 0) = x0

(5.48)

where

x =

2

64e�

v2D z✓

1

e�v2D z✓

2

✓3

3

75

T

2 H := L2(0, l)2 � R, x0

=

2

64e�

v2D z✓

10

e�v2D z✓

20

✓30

3

75

T

2 H

k1

(z) = � v2

4D� 2(k

1ss + k3

)yAss , and k2

= � v2

4D� k

2

The infinite dimensional representation of the system (5.47)-(5.48) on Hilbert space

H has the form of (5.9) by defining the operator A as:

A =

2

64Da

@2

@z2� k

1

(z) 0 �y2Ass

2k1ssyAss(z) Da

@2

@z2� k

2

y2Ass

0 0 ↵

3

75 (5.49)


dzare a.c.,

dx2

dz22 H,Da

dx1

dz|z=l = �v

2x1

|x=l

Dadx

2

dz|z=l = �v

2x2

|x=l, Dadx

1

dz|z=0

=v

2x1

|x=0

}(5.50)

and the boundary operator B by

Bx(.) =h

�Da@@z

+ v2

0 0i2

64x1

x2

x3

3

75

z=0

(5.51)

Assuming that, the control variable is x2

, the output operator C is:

C = C0

I =h0 1 0

i(5.52)

By performing the transformation (5.13), the operator A can be converted to a

block diagonal form and the decoupled infinite dimensional system (5.17) will be

computed. Using Equation (5.16), the operator J in Equation (5.13) is


J =

Z 1

0

T11

(t)A12

T22

(5.53)

T11

and T22

are the C0

-semigroups generated by �A11

and A22

. A11

,A22

and A12

are

given by

A11

=

"Da

@2

@z2� k

1

(z) 0

2k1ssyAss(z) Da

@2

@z2� k

2

#, A

12

=

"�y2Ass

y2Ass

#, A

22

=h↵

i(5.54)

• Computation of T11

A11

is a lower triangular operator and as discussed in §5.3 generates the C0

-

semigroup T11

given by

T11

(t) =

"T11

(t) 0

T21

(t) T22

(t)

#(5.55)

Where T11

and T22

are the semigroups generated by the diagonal elements of

�A11

and are given by

T11

(t)x =1X

n=1

e��nthx,�ni�n (5.56)

T22

(t)x =1X

n=1

e�µnthx, ni n (5.57)

�n, µn,�n and n can be calculated using the approach discussed in §5.2. T21

(t)

can be calculated using T11

(t) and T22

(t) by

T21

(t)x =

Z t

0

T22

(t � s)FT11

(s)xds (5.58)

where F is the o↵-diagonal element of A11

and is equal to 2k1ssyAss(z). By

performing simple calculations, T21

(t) becomes

T21

(t)x =1X

n=1

1X

m=1

�e��mt � e�µnt

�m � µm

hx,�mihF�m, ni n (5.59)


• Computation of T22

A22

is just an scalar and the semigroup generated by it is

T22

= e↵t (5.60)

• Computation of J

Using Equation (5.53) and Equations (5.55)-(5.60), and assuming that A12

="N

1

N2

#, J can be computed as:

J =

"J1

J2

#(5.61)

J1

x =1X

n=1

1

�n + ↵hN

1

x,�ni�n (5.62)

J2

x =1X

n=1

1X

m=1

1

(�n + ↵)(µm + ↵)hN

1

x,�nihF�n, mi m (5.63)

+1X

m=1

1

µm + ↵hN

2

x, mi m (5.64)

Finally, by defining A = ⇤A⇤�1, B = B⇤�1 and C = C⇤�1, the decoupled abstract

boundary control problem becomes:

8>>>>>><

>>>>>>:

dx(t)dt

= Ax(t)

x(0) = x0

Bx(t) = u(t)

y(t) = Cx(t)

(5.65)

The abstract boundary control problem (5.65), can be converted to a well-posed

infinite dimensional system with bounded input and output operators using Equations

(5.18)-(5.21). In the Equation (5.21), B can be calculated using the discussion in

Remark 5.1.2. Since B is any arbitrary function that satisfies conditions (5.22), we

assume that Bd =

"B

1

B2

#and B

1

and B2

are both second order polynomials. Using


the conditions (5.22), B1

and B2

are:

B1

=�2

4Dal + vl2z2 +

2

v(5.66)

B2

= � 1

4Dal + vl2z2 +

1

4Da + vlz +

2Da

4Dav + v2l(5.67)

Bl is any arbitrary number in R and we assume that Bl = 1. Finally B becomes:

B =

2

64B

1

+ J1

Bl

B2

+ J2

Bl

Bl

3

75 (5.68)

Now we are in a position to use the formulated LQ controller in §4.2 to control

the resulting well-posed infinite dimensional system of form (5.20) and analyze its

performance by numerical simulation.

5.6 Numerical Simulations

In this section the performance of the proposed approach is demonstrated. The LQ

controller discussed in the previous section was studied via a simulation that used a

nonlinear model of the reactor given in Equations (4.87)-(4.89) and (5.43). Values of

the model parameters are given in Table 5.1 (Weekman, 1969).

Table 5.1: Model ParametersParameter Value Unit

k1


k2

1.7 hr�1

k3


Da 0.5 m2hr�1

v 2 m⇥hr�1

yAin 0.7 weight fraction

yBin 0 weight fraction

↵ -0.001 hr�1

� 9.05 ⇥ 10�3

The control objective is to regulate the trajectory of yB at the desired steady state


profile. Deactivation of catalyst has a negative impact on yB and our objective is to

calculate the optimal values of inlet yA to keep trajectory of yB at the desired profile

and eliminate the e↵ect of deactivation. Using the nominal operating conditions,

and the model given in Equations (4.87)-(4.89) and (5.43), the steady-state profiles

of yA and yB were computed. Then, the nonlinear model was linearized around the

stationary states and transformed to the self-adjoint form of Equations (5.47)-(5.48).

Spectra of operators A11

and A22

were calculated using the algorithm discussed in

§5.2. In order to compute the spectrum of A11

, it was assumed that the length of the

reactor is divided into 50 equally-spaced sections and the coe�cient of the reaction

term is constant in each section. First five eigenvalues of the operator A11

are:

� = {�2.39 ⇥ 10�5,�1.34 ⇥ 10�4,�4.46 ⇥ 10�4,�1.12 ⇥ 10�3,�2.35 ⇥ 10�3}

The spectrum of A22

can be computed using Equations (4.41)-(4.43). First five

eigenvalues of A22

are:

� = {�2.04 ⇥ 10�6,�1.096 ⇥ 10�5,�5.68 ⇥ 10�5,�2.08 ⇥ 10�4,�5.78 ⇥ 10�4}

Finally, the spectrum of Ae was computed using Equations (5.33)-(5.35). The first

six eigenvalues of Ae is:

�(Ae) = {0,�0.001,�2.39⇥10�5,�2.04⇥10�6,�1.34⇥10�4,�1.096⇥10�5} (5.69)

and the associated eigenfunctions are shown in Figures 5.1-5.2. Once the eigenvalues

and eigenfunctions of the operator Ae are calculated, the LQ-feedback controller can

be computed using Equation (4.36). Note that since ⇧ is a self-adjoint operator,

h�n,⇧�mi = h�m,⇧�ni, therefore ⇧nm = ⇧mn. As a result, Equation (4.36) givesn(n+1)

2

coupled algebraic equations that should be solved simultaneously where n is

the number of modes that are used to formulate the controller. Since there are two

orders of magnitude di↵erence between the first and sixth eigenvalues of the operator

Ae, the e↵ect of higher order eigenvalues on the dynamic of the system is considered

to be negligible; therefore, in this work the first five modes were used for numerical


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1!1

!0.8

!0.6

!0.4

!0.2

0

0.2

0.4

0.6

0.8

1

z (m)

!n(2

)

n=0 n=1,3,5 n=2 n=4

Figure 5.1: Second element of �n given by Equation (4.67) and Equations (4.62)-(4.63)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1!4

!3

!2

!1

0

1

2

3

z (m)

!n(3

)

n=0 n=1 n=2 n=3 n=4 n=5

Figure 5.2: Third element of �n given by Equation (4.67) and Equations (4.62)-(4.63)

Sec. 5.7 Summary 111

simulation. The computed LQ controller was applied to the nonlinear model of

the reactor. Simulation of the nonlinear system was performed using COMSOL R�.

The closed loop trajectory of error is shown in Figure 5.3 and the optimal input

trajectory is shown in Figure 5.4. Figure 5.3 illustrates that, as catalyst deactivates,

the controller is able to regulate the trajectory of yB at the desired steady state

trajectory.

01000

20003000

40005000

0

0.5

1!0.02

0

0.02

0.04

0.06

0.08

time (s)z (m)

Err

or

0

0.01

0.02

0.03

0.04

0.05

0.06

Figure 5.3: Closed loop trajectory of error yB � yBss

5.7 Summary

The infinite dimensional LQ controller for boundary control of an infinite dimensional

system modelled by coupled Parabolic PDE-ODE equations was studied. This

chapter was a very important step in formulation of an optimal controller for the

most general form of distributed parameter systems consisting of coupled parabolic

and hyperbolic PDEs, as well as ODEs. The coupled PDE-ODEs were converted

to a decoupled form using an exact transformation. Conditions on existence of

Sec. 5.7 Summary 112

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000.68

0.685

0.69

0.695

0.7

0.705

t (s)

YA

in

Figure 5.4: Optimal input trajectory

this transformation were studied. The decoupled system was then formulated as

a well-posed infinite dimensional system by extension of the approach introduced

in Chapter 4. It was shown that the resulting system is a Reisz Spectral system

and using the properties of Riesz spectral systems, the stabilizability of the system

was proven. Since the transformed well-posed system was a Riesz Spectral system,

the LQ controller formulated in Chapter 4 was applicable to this system as well.

The LQ controller was applied to a catalytic fixed bed reactor, where the rate of

catalyst deactivation was modelled by an ODE. The closed loop performance of

the controller was studied via numerical simulations. It was illustrated that the

formulated controller is able to eliminate the e↵ect of the catalyst deactivation.

6Conclusions and Recommendations

This thesis was concerned with the problem of the infinite dimensional optimal

controller design for distributed parameter systems with special emphasis on

transport-reaction processes (i.e., fixed-bed reactors) that are normally modelled

by partial di↵erential equations. Since the governing transport phenomenon in a

transport-reaction system determines the type of the PDEs involved in the modelling

(e.g., hyperbolic or parabolic), this thesis was divided to two parts to address

each type of PDE system independently. The goal was to develop the required

mathematical tools for solution of the optimal control problem for a general class of

distributed parameter systems representing a fixed-bed reactor with minimum number

of simplifying assumptions in the modelling of the reactor.

The first part of the thesis concentrated on catalytic reactors with negligible

di↵usion that are modelled by a set of hyperbolic systems. In Chapter 2, a LQ

controller for a set of time-varying coupled hyperbolic equations was formulated.

Exponential stability of these systems was analyzed using the Lyapunov approach. It

113

114

was shown that the solution of the optimal control problem could be found by solving

an equivalent matrix Riccati partial di↵erential equation. Numerical simulations

were performed to evaluate the closed loop performance of the designed controller

on a hydrotreating fixed-bed reactor. The performance of the proposed controller

was compared to the performance of an infinite dimensional controller formulated by

ignoring the catalyst deactivation. Simulation results showed that the performance of

the proposed controller is better than the controller that ignores catalyst deactivation,

when the deactivation time is comparable to the residence time of the reactor. In case

of very slow deactivation the two controllers had comparable performance.

Input and output constraints, as well as parameter uncertainty, are issues that

shall be considered in designing a controller. Chapter 3 addressed the problem

of constrained model predictive control of hyperbolic systems under parameter

uncertainty. A characteristic based robust model predictive control algorithm was

developed for a set of two time-scale hyperbolic PDEs. This class of hyperbolic

systems have two characteristic curves along which the set of PDEs can be represented

as a set of ODEs. Using this feature, the set of partial di↵erential equations was

transformed to a set of ODEs. In order to construct the state space model of the

system, the spatial domain was discretized by a finite number of discretization points.

The structure of the resulting state space model was explored. It was shown that the

resulting state space system has certain periodic features depending on the ratio of the

slopes of two characteristic curves. A very nice feature of the proposed algorithm is

preservation of finite impulse response property of the co-current hyperbolic systems.

This means the proposed method can tolerate discontinuities in the boundary

conditions as opposed to other discretization methods. In order to specify the

uncertainty of the plant, it was assumed that the real model of the system lies inside a

polytopic set ⌦ generated by uncertain models of the system. The MPC problem was

defined by nominal performance index but constraint satisfaction for all polytopic

models was ensured. A case study involving a hydrotreating reactor was used to

evaluate the closed loop response of the controller. In this case study two uncertain

115

parameters leading to four polytopic models existed. Simulation results indicated

that this algorithm is able to satisfy input and output constraints under parameter

uncertainty.

The second part of the thesis dealt with systems modelled by a set of parabolic

PDES. In chapter 4, optimal boundary control of infinite dimensional systems

described by a set of parabolic PDEs with spatially varying coe�cients was studied.

Such systems are results of linearization of nonlinear PDE model of the process around

its steady state profile. Using an exact transformation, the boundary control problem

was transformed to a well-posed infinite dimensional system. It was proven that

the resulting system is a Riesz Spectral system. Stabilizability of the system was

investigated using the spectral properties of the system. The most challenging part

of this chapter was to solve the eigenvalue problem for the set of coupled PDEs with

spatially varying coe�cients. The spectrum of the system is required to solve the

optimal control problem for Riesz-Spectral systems and unfortunately there is no

method available for analytical solution of a general form of the parabolic operator.

Therefore, some simplifying assumptions were made and then an innovative approach

for solution of eigenvalue problem was proposed. The basic idea for solution of the

problem was the solution method for the heat equation in composite media. Moreover,

by using the spectral properties of the system, the operator Riccati equation was

converted to a set of algebraic equations that could be solved numerically.

A tubular reactor with axial dispersion was considered as the case study. The

reactor can be modelled by a set of nonlinear parabolic partial di↵erential equations.

Linearization around the steady state profile of the system results in a set of

linear PDEs with spatially varying coe�cients. The performance of the formulated

controller was compared to a LQ controller based on finite di↵erence algorithm.

Simulation results showed that the infinite dimensional controller leads to better

performance in terms of l2-norm of the error.

In Chapter 5, the tools developed for optimal control of parabolic systems in

Chapter 4 were extended to the boundary control of an infinite dimensional system

Sec. 6.1 Future Work 116

modelled by coupled Parabolic PDE-ODE equations. An exact transformation was

introduced to convert the system to a decoupled form. Conditions on existence of

such a transformation were studied. Using another transformation, the boundary

control problem was formulated as a well-posed infinite dimensional system. It was

shown that the resulting system is a Reisz Spectral system and using the properties

of the Riesz spectral systems, the stabilizability of the system was proven. Since

the transformed well-posed system was a Riesz Spectral system, the LQ controller

formulated in Chapter 4 could be applied to it. This chapter took a very important

step in controller formulation for more general forms of distributed parameter systems

consisting of coupled parabolic and hyperbolic PDEs and ODES.

6.1 Future Work

This thesis took a significant step towards building required tools for solution of

optimal control problem for a general class of distributed parameter systems with

special focus on catalytic reactors. A number of challenges remain in the development

of the infinite dimensional controller for processes modelled by a general form of

infinite dimensional systems.

The Robust MPC studied in Chapter 3 used the method of characteristics as a

tool to convert the set of hyperbolic equations to state space system. This method

performs very well for systems with up to two characteristic equations, but for systems

with more than two characteristic equations the computation of the state space model

becomes very complicated. The method of characteristics is a special case of invariant

transformations in mathematics. For general case of hyperbolic system, an invariant

transformation approach might be useful in conversion of PDEs to ODEs. This

approach has not been used in engineering and could lead to significant contributions

in the area of infinite dimensional controllers.

The method proposed in Chapters 4 and 5 had a limitation on the structure of the

operator A. In order to solve the eigenvalue problem it was assumed that the infinite

dimensional operator had a triangular form, which means the coupling of the state

Sec. 6.1 Future Work 117

variables is one way. Solution of the eigenvalue problem for general form of infinite

dimensional operators cannot be performed analytically. For special cases the infinite

dimensional operator can be triangularized using a transformation. Exploration of

the conditions for existence of such a transformation is the topic that needs to be

addressed in future work.

A final important area for future work is to formulate infinite dimensional

controllers for systems modelled by a set of hyperbolic-parabolic PDEs. Many

chemical processes heat transfer can be represented by a hyperbolic PDE, but mass

transfer is modelled by a set of parabolic PDEs. Since the approaches for solution of

these two types of PDEs are di↵erent, it would be worthwhile to investigate how to

exploit two di↵erent approaches within one problem.

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Documents

University of Alberta · controllers using accurate models of the processes. Modelling and control of ﬁxed-bed reactors have been the topic of many researches since early 70’s